Y-chromosomal Aaron
Y-chromosomal Aaron is the name given to the hypothesised most recent common ancestor of many of the patrilineal Jewish priestly caste known as Kohanim (singular "Kohen", "Cohen", or Kohane).  In the Hebrew Bible this ancestor is identified as Aaron, the brother of Moses.
Research published in 1997 and thereafter has indicated that a large proportion of contemporary Jewish Kohanim share a set of Y chromosomal genetic markers, known as the Cohen Modal Haplotype, which may well derive from this single common ancestor.

# Background
Although membership in the Jewish community is traditionally passed maternally (see: Who is a Jew?), membership in the group that originally comprised the Jewish priesthood ("Kohens" or Kohanim), is patrilineal, and modern Kohens claim descent from Aaron, brother of Moses.
For human beings the normal number of chromosomes is 46, of which 23 are inherited from each parent. Two chromosomes, the X chromosome and Y chromosome, determine gender. Women have two X chromosomes, one inherited from their mother, and one inherited from their father. Men have an X chromosome inherited from their mother, and a Y chromosome inherited from their father.
Males who share a common patrilineal ancestor should also share a Y chromosome, diverging only with respect to accumulated mutations.  Since Y-chromosomes are passed from father to son, all Kohanim men should theoretically have almost identical Y chromosomes; this can be tested with a genealogical DNA test.  As the rate that mutations accumulate on the Y chromosome is relatively constant, scientists can estimate the elapsed time since two men had a common ancestor.  (See molecular clock.)

# Initial studies
The Cohen hypothesis was first tested by Prof. Karl Skorecki and collaborators from Haifa, Israel, in 1997.  In their study, "Y chromosomes of Jewish priests," published in the journal Nature, they found that the Kohanim appeared to share a different probability distribution compared to the rest of the Jewish population for the two Y-chromosome markers they tested (YAP and DYS 19); and that furthermore the probabilities appeared to be shared by both Sephardi and Ashkenazi Cohens, pointing to a common Cohen population origin before the Jewish diaspora at the time of the Roman empire.
A subsequent study the next year (Thomas MG et al, 1998) increased the number of Y-STR markers tested to six, as well as testing more SNP markers.  Again, they found that a clear difference was observable between the Cohanim population and the general Jewish population, with many of the Cohen STR results clustered around a single pattern they named the Cohen Modal Haplotype:
Here, becoming increasingly specific, xDE is the proportion who were not in Haplogroups D or E (from the original paper); xDE,PR is the proportion who were not in haplogroups D, E, P, Q or R; Hg J is the proportion who were in Haplogroup J (from the slightly larger panel studied by Behar et al (2003)); CMH.1 means "within one marker of the CMH-6"; and CMH is the proportion with a 6/6 match.  The final two columns show the conditional proportions for CMH.1 and CMH, given membership of Haplogroup J.
The data shows that the Cohanim were more than twice as likely to belong to Haplogroup J than the average non-Cohen Jew; and of those who did belong to Haplogroup J, the Cohanim were more than twice as likely to have an STR pattern close to the CMH-6, suggesting a much more recent common ancestry for most of them compared to an average non-Cohen Jew of Haplogroup J.
Thomas et al dated the origin of the shared DNA to approximately 3,000 years ago (with variance arising from different generation lengths).  The techniques used to find Y-chromosomal Aaron were first popularized in relation to the search for the patrilineal ancestor of all contemporary living humans, Y-chromosomal Adam.

## Responses
The finding led to excitement in religious circles, with some seeing it as providing some "proof" of the historical veracity of the Bible  or other religious convictions, but there was also criticism that the paper's evidence was being overstated.

# Cohens in other haplogroups
Behar's 2003 data points to the following Haplogroup distribution for Cohens as a whole:
The detailed breakdown by 6-marker haplotype (the paper's online-only table B) suggests that some at least even of these groups (eg E3b, R1b) contain more than one distinct Cohen lineage.  It is possible that still further other lineages may also exist, but were not captured in the sample.

# Does a CMH prove Cohen ancestry?
One source of early confusion was a widespread popular notion that only Cohens or only Jews could have the Cohen Modal Haplotype.  It is now clear that this is not the case.   The Cohen Modal Haplotype, whilst notably frequent amongst Cohens, is also far from unusual in the general populations of haplogroups J1 and J2 with no particular link to the Cohen ancestry. These haplogroups occur widely throughout the Middle East and beyond ,.  So whilst many Cohens have haplotypes close to the CMH, a far larger number of such haplotypes worldwide belong to people with no likely Cohen connection at all.
Statistically the value of matching the CMH can be assessed using Bayes' theorem, which in its odds form can be written:
In words, this says that the odds in favour of Cohen ancestry C (ie the probability of having Cohen ancestry, divided by the probability of not having Cohen ancestry), having observed some piece of data D, is given by the odds one would assign given only one's initial information I, multiplied by the probability of having observed D if C is true, divided by the probability of having observed D if C is false.
(In fact, for convenience we shall work with the reciprocal of this equation, ie work in terms of odds against, rather than odds on).
The proportion of the whole male Jewish population that has Cohen ancestry has been estimated at 5%.  So if we take that 5% as our initial estimate of the probability of shared Cohen ancestry, then on the basis of the data above:
- Not belonging to haplogroups D or E  improves the odds for a Sephardi Jew from 19/1 against to  (19/1)*(0.85/1.00) = 16.2/1 against (a 5.8% probability)
- Not belonging to haplogroups D,E,P,Q or R takes the odds to (19/1)*(0.63/0.88) = 13.6/1 against (6.8% probability).
- Membership of Haplogroup J improves the odds to (19/1)*(0.37/0.75) = 9.4/1 against (9.6% probability).
- Being within the CMH.1 group takes the odds to (19/1)*(0.14/0.61) = 4.4/1 against (18.7% probability).
- A full 6/6 match takes the odds to (19/1)*(0.10/0.56) = 3.4/1. (22.7% probability).
Even a full 6/6 match for the 6 marker CMH thus cannot "prove" Cohen ancestry.  It can only somewhat strengthen a previously existing belief.  But for populations where the background probability assessment of shared Cohen ancestry must be vanishingly low, such as almost all non-Jews, even a full 6/6 match makes only a small difference.  For individuals in such populations the CMH likely indicates Haplogroup J, but a completely different ancestry to the Cohanim.

# Higher resolution
The discussion above applies to the so far published scientific papers.  However, in principle some more resolution could be obtained by determining the Cohen haplogroup more narrowly, and/or testing more Y-STR markers to determine whether there is an extended characteristic Cohen haplotype.

## Haplogroup placement
The largest population of Kohanim which most closely match the Cohen haplotype cluster are believed to belong to subgroup J1 of haplogroup J.
Individuals with the genetic Cohen Modal Haplotype can be found in subgroup J2 as well, and occasionally in more genealogically distant haplogroups too; however these are not closely related to the cluster in Haplogroup J1.
The subdivision of J2 which most closely matches the genetic signature of the J1 Cohens is subclade J2a1b, a large fraction of members of which will also have a 6/6 match for the 6-marker CMH.  However, this is an example of (re)convergence of haplotypes of different genetic lines, which it is believed have been not been closely related for at least the last 10,000 years; the group in J2a1b who have the 6-marker CMH are devoid of any Cohen traditions in their families.
On the other hand, there are families in Haplogroup J2 who do have a Cohen religious tradition and are proud of it (as there are in several other haplogroups, including Haplogroup R1b).  The haplotypes of these Haplotype J2 Kohanim cluster in a unique, small offshoot of J2a1*, close to haplotypes of the J2a1k clade, not the J2a1b clade.  These J2 Kohanim typically have a 4/6 match for the 6-marker CMH (with DYS19=15 rather than 14, and DYS388=15 rather than 16).  They do not match the 12-marker J1-extended CMH, and they do not share a common ancestor with the J1 Kohanim in a Biblical timeframe; but they are equal co-inheritors of a patrilineal tradition which appears to date back well before the Diaspora.
As it happens, three of the four markers for which they do match the CMH-6 were the markers tested by Malaspina et al (2001).   This appears to explain the finding of that paper that "typing a limited number of Italian Cohanim (A. Novelletto unpublished obs.) for the STRs used here, we determined that the Cohen Modal Haplotype ('an important component in the sharing of Ashkenazic and Sephardic Israelite Y chromosomes', Thomas et al. 2000) does indeed belong to network 1.2" (ie the population having DYS413a,b<=18, which is the signature of the J2a1 subclades).

## More detailed Cohen haplotypes
In the table below, the first line gives the original 6 marker Cohen Modal Haplotype (CMH-6), which was the basis for the original published papers.  The second gives an extended 12 marker haplotype (CMH-12) informally released by the private company FTDNA, based on further work by much of the same research team.  It has not yet been peer group reviewed by other scientists or published in the open technical literature.
The next sequence of rows identify other 6-marker haplotypes in haplogroup J found to occur more than once in the sample of 145 Cohanim tested in Behar et al (2003)  (table B (web-only) in that paper).  Probable extensions of these haplotypes to 12 markers are shown, where it has been possible to find corresponding clusters of Cohen-type names in publicly accessible DNA databases, together with the apparent sub-clade of haplogroup J.  This is more possible for the apparently Ashkenazi clusters than for Sephardis, who are much less strongly represented in the databases.
Finally, for comparison, the 12-marker modal haplotypes for the haplogroups J1 and J2 are also shown.  It is apparent that in both cases, their haplotype clusters are also centred very close to the Cohen modal haplotype.  However, because of the much greater time that has elapsed since the mutations occurred that define the haplogroups, there has been much more time for Y-STR mutations to build up; so, although they have almost the same centre as the Cohen cluster, the J1 and J2 haplotype clusters are much more diffusely spread out.  Thus although the CMH-6 is also very near to the most probable haplotype for both J1 and J2, its occurrence frequency is only about 1 to 8% amongst arbitrary members of haplogroup J with no particular Cohen connection.

# Other carriers of the DNA
Critics of the theory point out that the Cohen Modal Haplotype has also been found in significant numbers in groups of non-Jews, notably Italians.  However proponents of the theory are quick to explain these anomalies.  They state that history records the migration of large numbers of Jews to Italy who married and took part in building the Colosseum.
However, the Cohen Haplotype has also been found among significant numbers of non-Jewish Arab and Kurdish populations.  Although this too could be explained in part by miscegenation, assimilation, and conversions, there is an alternate possibility that the marker precedes all of these populations and is a precursor to some common ancestral group.

## Lemba
The Cohen Modal Haplotype has also been found in the Lemba of Southern Africa, who have a tradition of Jewish ancestry (Thomas MG et al 2000).

## Kurds
The Cohen Modal Haplotype has also been found in some groups of Kurds.  There are a number of Kurdish Jews.

# Y-chromosomal Levi?
A similar investigation was made with men who consider themselves Levites. Whereas the priestly Kohanim are considered descendants of Aaron, who in turn was a descendant of Levi, son of Jacob, the Levites (a lower rank of the Temple) are considered descendants of Levi through other lineages. Levites should also therefore share common Y-chromosomal DNA.
The investigation of Levites found high frequencies of multiple distinct markers, suggestive of multiple origins for the majority of non-Aaronid Levite families.  One marker, however, present in more than 50% of Eastern European (Ashkenazi) Jewish Levites points to a common male ancestor or very few male ancestors within the last 2000 years for many Levites of the Ashkenazi community. This common ancestor belonged to the haplogroup R1a1 which is typical of Eastern Europeans, rather than the haplogroup J of the Cohen modal haplotype, and most likely lived at the time of the Ashkenazi settlement in Eastern Europe. .
The E3b1 haplogroup has been observed in all Jewish groups world wide. 
It is considered to be the second most prevalent haplogroup among the Jewish population. According to one major paper, .
It has also been observed in moderate numbers among individuals from Ashkenazi, Sephardic and Samaritan backgrounds that contain the E3b1 haplogroup, having a tradition of descending from the tribe of Levi, suggesting that the E3b1 Levites may have existed in Israel before the Diaspora of 70 C.E.
The Samaritan community is a small, isolated, and highly endogamous group today numbering some 650 members who have maintained extensive genealogical records for the past 1315 generations. Since the Samaritans maintain extensive and detailed genealogical records, it is possible to construct accurate pedigrees and specific maternal and paternal lineages. The Samaritan community in the Middle East survives as a distinct religious and cultural sect and constitutes one of the oldest and smallest ethnic minorities in the world. Y-Chromosome studies have shown that the majority of Samaritans belong to haplogroups J1 and J2 while the Samaritan Cohanim belong to haplogroup E3b1a.. In 1623-1624 the last member of the high-priestly family, which claimed descent from the eldest son of Aaron, died. The office was then given to the junior branch, descended from Uzziel, the son of Kohath. Since that date the priest has called himself "ha-kohen ha-Lewi," (Heb. "The Levite Priest") instead of "ha-kohen ha-gadol" (Heb. "The High Priest") as in previous times. The approximately 650 individuals comprising the total group of present day Samaritans trace their ancestry over a period of more than 2,000 years to the Biblical Israelite tribes of Ephraim, Menashe and Levi. As a religious sect, the Samaritans broke away from the main stream of Judaism around the fifth century B.C.E.