For my work, I'm working on the impact of the GDPR on the research, and how will the GDPR work in scientific communities. I'm not a lawyer, of course, so my interpretation might be a bit off (so disclaimer, IANAL, this is not a legal advice, and etc.). Anyway, these are just some of my thoughts on the subject.

Well, GDPR is a big topic, and it not yet clear how all the provisions will be implemented. It is not that different from the (currently valid) Directive, but it does clarify certain points, and makes much more stringent penalties, as mentioned in parent post (the fine is actually 4% of the global revenue, or 20M Euro, whichever is greater).
The changes in respect to the Directive are, in short:

• GDPR applies to the processing of personal data by controllers and processors in the EU, regardless
where it takes place

• Penalties – up to 4% of annual global turnover or 20M€ (whichever is greater)

• Consent – conditions are strengthened (clear and plain language, explicitly related to the
processing, easy to withdraw)

• Breach notification

• Privacy by design

• Right to be forgotten

• Data Protection Officers

• Right to access

Now, as mentioned in another comment, the right to be forgotten and erasure of data is not really wipeout, the data controller and data processor are supposed to do it using "industry standards" and "reasonable effort" (controller, e.g. should flag that the processing the data should be restricted). Also, there are exceptions (legal claims, public authorities, free speech, etc.).

Different comment points out that the Regulation, unlike Directive, makes GDPR valid in all EU countries, and this is true. However, the EU states are free to implement their own data privacy laws, which of course, need to be in line with the GDRP. This may potentially introduce legal inconsistencies across the EU for certain points.

Also, one should not underestimate the legitimate interest of the service provider, or controller, to retain the data, even if the user has asked for the data to be removed. The data may also be retained by the request of relevant public authorities, etc. One comment has suggested what will happen if the EU citizen requests the removal of it's data, while the US public authorities asks for access to this data. In this case, the relevant EU public authorities may request for the data to be kept (or not, I guess this will be decided on case by case, also the provider may have a legitimate reason to keep the data..).

And of course, the biggest problem, the transfer of data to non-EU countries. For this, there are several ways to do it, one is mentioned already, i.e. user consent (which must be clear and unambiguously given, and can be revoked at any time). Then, of course, there are contracts, binding corporate rules, etc. For EU-US transfer, there is Privacy Shield for transfer of data to US (which is a replacement for the Safe Harbor, stricken by EJC), but this is mostly for commercial services (so it does not work for academic environments..).

There are some other interesting aspects to GDPR, but this post is already getting a bit long. For more info, these links are interesting:

[1] https://aarc-project.eu/aarc-infoshare/ -- for academic environments..

[2] https://iapp.org/resources/article/top-10-operational-impact...

[3] https://www.whitecase.com/publications/article/unlocking-eu-...

There are multiple WP29 interpretations on various points (some of them are actually human readable, not just legal talk..), etc. In any case, it will be interesting to see all these developments in the future.

[Edited for mistakes..]