10 common themes between the advice below:

Hal Abelson
I’m looking for leadership and initiative. My group has a large number of undergraduate students and I look to our grad students to be role models and leaders. Compelling application essays should talk about actual accomplishments: applications you’ve created that others are using, technical organizations you’ve started or where you play a major role. There of course needs to be a track record of academic excellence. But the centerpiece of my group is empowering people of all ages through technology, as users and creators. That passion should come through in the essay–and it’s even better if there’s a track record to point to.

Karl Berggren
An application essay provides a number of useful information points when I’m reading a candidate’s application. I’m mostly looking to see if the person can communicate clearly. Second, I’m trying to find out a bit about the person, especially their personality and motivation, and how they think about science and engineering. Finally, I’m interested in learning a bit about what circumstances have shaped the candidate’s life. This is the place where I want to know if someone has faced exceptional challenges or took advantage of a unique opportunity to do something special. Because of the importance of writing in an academic environment, I’m looking to see if there is structure to the essay, and if paragraphs are well formed. For me, the essay is really not usually the main compelling reason to admit a student, but an essay that is over the top, or is poorly written or poorly structured, sometimes puts me off a candidate who otherwise would be a possible admit.

Adam Chlipala
I think it’s useful to think of PhD applications as more like job applications than earlier educational applications. You are applying to be an apprentice researcher, and thus concrete research experience (attested to by knowledgeable supervisors in letters) is most useful to give evidence that you will succeed. Then the specifics vary by research area. I’m looking for specific technical skills and bigger-picture direction-setting skills. In my area, the former are a mix of software/hardware implementation at a decently large scale and comfort with rigorous math and logic. The latter come down to finding ways that computer systems are developed ineffectively today, and thinking up ways we could change the development tools–ideally applicants can point to cases where they were the ones driving that brainstorming, not just implementing ideas coming from supervisors. Giving some examples of project directions you want to explore is helpful both for showing that kind of initiative, and for helping potential advisors gauge fit with their own interests. However, don’t worry that anyone will hold you to working on any of the specific ideas you list!

Frédo Durand:
Mostly two things:

1/ Can the applicant think and write deeply and intelligently about state-of-the-art technical issues?

2/ What kind of technical area (very broadly speaking) are they interested in?

One challenge for applicants is that the two answers sometimes conflict: the first question requires narrowness, but you probably want to show you’re interested in a broader set of topics to maximize the number of faculty members who feel they may want to work with you. So, I usually advise students not to restrict their essay to their past research, but have a paragraph or two at the end or beginning to list the areas that appeal to them. Ideally, the letter would give me a sense of how they attack an open problem, through the example of past projects (ideally research projects).

Regarding recommendation letters, I want to read about a candidate’s potential to do research. The most insightful letters are from people who have supervised you on a research project, or on a project that involves a fair amount of independence and creativity. I find letters from people who only know a candidate in a class context less useful, unless the student has done a particularly remarkable project. Letters from mentors in industry can be useful, especially if they do or have done research. However, not all applicants (including successful ones) have three letters that are equally thorough, and it’s quite common to have 1 or 2 letters from people who really know you well from a research perspective, and 1 or 2 that are a little more superficial.

Polina Golland
The essay should focus on your interests and look to the future. Describe what problem you would like to tackle in the future and approaches you might want to take. Even if I disagree with what the applicants write, it is revealing on how they think and gets me interested. Keep the description of your (very impressive) past projects to the minimum, mention them only as support for what you want to do in the future. Your CV, other sections of the applications, and recommendation letters will talk about past projects, and it’s a pity to use your essay to rehash it again.

Piotr Indyk
For me, the most important aspect of the application is the evidence of research skills. In the ideal case, it would take the form of publications/manuscripts describing the research project(s) and results, which I can read and review. Descriptions of research projects in recommendation letters and/or the research statement are less optimal, but also OK. However, it is understandable that not all applicants have the same opportunities to pursue undergraduate research. In such cases, I try to infer from other parts of the applications, such as grades (esp. for relevant technical subjects), recommendation letters and other activities like olympiad participation. Regarding the research statement, I find it to be useful as a broad indicator of applicant research interests, but since interests of many (most?) applicants evolve, I do not put that much weight on it.

Daniel Jackson
When I think about taking on a graduate student, I ask myself: is this a good match? I want to be sure that we’ll enjoy each other’s company and be successful working together. That means you’ll be excited by the kind of work I do, and have a reason to think that working with me will be better for you than working with someone else. The students I tend to gel most with want to reconsider how we design and build software, and like to think deeply (and even a bit philosophically) about the fundamental problems. So I read the statement carefully, looking for someone who thinks clearly and creatively, knows a little bit about what I do, isn’t too distracted by technology or formalism for its own sake, and is eager to pursue big ideas. And if there’s some project you’ve done that shows some promise (especially if one of your letter writers can talk about it), so much the better.

Leslie Kaelbling
For me, the most important thing in an application is the best letter of recommendation, by a large margin. The main research letter should speak to the candidate’s creativity, independence, bravery, and ability to get things done. The other letters usually don’t matter much. I like unusual candidates, and am generally more interested in someone who has done something on their own, or in an unusual place, than someone with a lot of papers who spent four years in a very productive and prominent research group. Most essays are neither a positive nor a negative for the application. The ones I remember and value are ones that I learned something from–essays that are actually interesting to read because they have a strong or novel view or that articulate a clear vision. I also like to get the feeling that the candidate really values research intrinsically–that they are not simply applying to do a PhD because it seems like a good stepping stone to something else, or something that is highly esteemed by others. I don’t worry about a few poor grades, if they have an explanation: early in the student’s career, or during one rough semester, or as the result of exploration. I’d rather see a student with a few Bs or a C, who has taken challenging classes, than someone with a perfect GPA and completely standard undergraduate curriculum. I am completely unimpressed by a student who takes twice the normal course load—they should have been doing research!

David Karger
There are two questions that I ask when I’m looking at an application.

(1) Will this student be interested in working on the kinds of things that excite me?

And (2) do they have the independence and organization necessary to work for a laid back, disorganized advisor like me?

For the first question, I like to hear what specific problems interest you, and why. Not why you’ve loved computer science since you got your first PC at age 4, but why you consider certain specific problems important and interesting, and how you might go about trying to solve them. It certainly doesn’t hurt to have looked at some of the work my group is doing, talk about why it’s interesting to you (not just that it is), and maybe give some thoughts on where it might be interesting to take it further. But hearing your own ideas is wonderful too. You need not have solved them already, although it is great to talk about a problem you have already worked on.

The second question is specific to my advising style. I provide a lot of support and feedback to my students, but I don’t do a lot of management. So it’s important for me to know that a student will take initiative, make choices about what to work on, make decisions about designs and implementations, set their own deadlines and meet them, and come to meetings with ideas and questions to move the work forward. Just claiming this in your statement isn’t particularly meaningful, but I look for signs of it in past work (and recommendation letters).

Manolis Kellis
I’m looking to see several things:

(1) Clarity of thought: this comes through in the essay; the vision they have (for the field, how their works fits in, the broader perspective within, next steps, etc); the way they describe their accomplishments (organization, background, clarity of innovation, are they able to explain the problem, the challenge, the novelty, etc); and, of course, their grades and accomplishments.

(2) Research accomplishment: Show that they can innovate, invent, find problems, frame them, and bring things to completion, writing papers, completing projects, packaging up code, creating tools.

(3) Letters: Evidence of standing out, innovation, novelty, ability to make progress independently, yet team spirit and collaboration.

(4) Technical: Of course, their training, the rigor, the background, grades, competitions, etc.

(5) Passion: Especially for an applied field like genomics/biology/medicine, showing that they truly care about the application area, not just about the algorithms, but that they truly have sought to find something novel in the specific application area that they have chosen, and been able to interpret their results and make conclusions about the applied field.

Stefanie Mueller
For me, the most important is that the applicant can show that they have research experience in my research field. Hiring a PhD student is a 5-6 year commitment, so it is very important for me that the applicant can show me that we will produce exciting research together. When I read an application, I first check if the applicant has publications in my research field on topics related to what my research group works on. After this, I look at the recommendation letter writers and see if they come from faculty in my research field and if they talk about that the applicant can conduct research in my field. Letters from outside my research field are not very useful in determining if the applicant can do research in my area. After this, I read the statement of purpose to see if the applicant has ideas that I would also be excited to work on.

Will Oliver
When I read a graduate student applicant’s research statement, I look to obtain a picture of the student and their research interests. This includes the student’s motivations for research. Who is this person, where did they come from, what sparks their interest in science and engineering, how has that been reflected in their lives and their trajectory? I then look for examples of research experience, broadly defined. This could be an experience as an undergraduate researcher, a summer internship, or even a substantial hobby project (to name a few). I look for tangible outputs from those projects, such as a peer-reviewed publication. I then look for what the student wants to accomplish in graduate school. I appreciate a genuine exposition of intellectual curiosity and enthusiasm in describing these goals. While this approach naturally leads to some specificity in research topics, I also look for some flexibility and breadth. For example, even if the student has one top-choice topic or group, it is a good idea to articulate other (often related) areas that would also be of interest.

Al Oppenheim
A long time ago one of my graduate students asked me what I look for in choosing the students to work with. My quick and somewhat playful answer was that I have four criteria: intelligent, creative out of the box thinker, enjoyable to interact with, and coachable. From applications on paper and without personally meeting the applicant, it’s often hard to assess these and particularly the second, third, and fourth. An approximation to the first can be based somewhat on the transcript. The other three, perhaps mostly from the reference letters and personal statement and when possible personal interaction. The two areas in which applications are often weak are in the choice of references and the lack of detail in the reference letters, and in the crafting of the personal statement. In writing the personal statement I’ve typically advised potential applicants to use it as an opportunity to truly show their motivations, goals and personality, rather than trying to pattern match to what they think readers will give high marks to. If the personal statement is genuine and honest, it shows. And if it isn’t, it also shows.

Gerald Jay Sussman
The problem is that we have too many “excellent” applicants, most of whom would do fine in our graduate program. Most would do good, publishable, but incremental research. We accept plenty of those excellent people. But I am looking for the candidates that could break a paradigm and open up a new field of research.

So what I look for in an application is evidence that the candidate has an unusual perspective, perhaps in conflict with the conventional wisdom of the field. I am open to considering crackpots, but I also look for evidence of technical skill and clarity of thought and expression that separates the interesting characters from crackpots.

Additionally, the most persuasive information in an application is reference letters from previous supervisors or teachers who attest to the skill and creativity of the applicant.

George Verghese
Most of my reading of graduate folders necessarily happens at the initial stage of evaluations, when I’m looking for applicants who seem like they would thrive in, and contribute strongly to, our department or a broad research area within it. The application folders that emerge from this reading then get passed on to other faculty for more focused evaluation. So at the initial stage I am not necessarily looking for a good match to my personal research interests or style; that can come later, when I look at short-listed folders, perhaps sent my way by other faculty who think I might want to take a look.

For the initial reading, I first examine the applicant’s academic record, to be reassured that they will be able to handle at least the course work in our graduate program. A few blemishes in early years may be fine, but anything less than a strong overall academic record is likely to be a non-starter (though I will read quickly all the way through the application, feeling I owe at least that much to a hopeful applicant who has paid their application fee!).

I then look for tangible, documented outcomes of activities that go beyond standard academic efforts, whether unusual and independent projects (not standard lab projects in a class), or in research or internships. An important part of the substantiation is in the letters of reference, which have to reflect genuine, specific, modulated knowledge of the applicant and their work, and corresponding enthusiasm. A letter that sounds generic, though filled with superlatives and rating the candidate as Truly Exceptional, will not count for much. I want to know that the letter writers see stellar achievement (in academics and beyond) and potential, based on the specifics of their interaction with and knowledge of the applicant.

Finally, I turn to the student’s statement to get a sense of their voice, how they see and present themselves and their accomplishments, and what they’re looking to find in/with their graduate work. A well-crafted statement that comes across as mature, genuine, and reasonably aware of the field in which they hope to concentrate counts for a lot.

Ryan Williams
When reading a grad school application, I focus on several things. The first (obvious) thing is whether my interests align well with the applicant. This doesn’t necessarily mean that the applicant is working on the same exact problems as me; it means that I try to understand their taste in problems and topics, and how this fits with what we study in my research group. Another important thing is independence (in research, thought, etc): we get many applications from many talented students from all over the world, but we don’t see too many who showed a significant degree of independence in their thought and behavior, different from those around them (including their mentors). This can come out in the statement of purpose, but often more so in the letters. “Independence” can have various interpretations, and I’m deliberately leaving the term somewhat vague, because I think any of those interpretations can be important. Another important thing is the quality of their communication, especially their writing. I always read the statements of purpose carefully, as well as any writing samples the applicant has provided. Of course, letters of recommendation which attest to all of these qualities are also very helpful.