How to take notes in PDFs in Linux

Xournal can be used to annotate PDFs (and with proper text, not just like the rather whimsical screenshot).

I've used it to fill in non-interactive PDF forms.

If you are using KDE, you can try Okular, the standard document viewer of this desktop environment. When in review mode, it can add a large number of highlights and annotations to PDF files

The Foxit Reader is also available in a native version for Linux and offers a plethora of annotation features.

Software: https://www.foxitsoftware.com/products/pdf-reader/

I use pdfescape and edit my pdfs online. The comments are saved to the original pdf. http://www.pdfescape.com/

Update: Master PDF editor, which is free for linux (non-commercial use), is also a good option. I have been using this for almost a year now, and it suits most of my needs.

Mendeley can do all of this, plus it is an awesome software for keeping track of your PDF:s (with searching, tagging etc). Also, PDF XChange Viewer is a free windows program that can do this too, and works well under Wine (it's even very fast). I have used both of these for taking notes during my engineering studies.