Is there a database of chess problems that have the game or positions PGN and or FEN?
Some large chess puzzle databases with links:
Free open source collection of 1,743,871 Problems at chessblunders.org, collected from more than 6 million games from professional tournaments.. They have an API for access to database.
Taken from the website "wtharvey":
From the website "the art of mate in 2"
- Nearly ~152000 puzzles in pgn format
- You will also find four pgn files of ~ 500 puzzles made by "Argentine problemists"
Browsable collection of problems at "Yet another chess problems database". There are ~420000 problems. Not sure whether it can be downloaded programmatically. In the "About" page, you will find more links, some of which are broken.
You can also buy Harold Van der Heidjen's endgame study database. It contains more or less all the endgame studies he could gather from the old classic (Saavedra, Ponziani etc..) to the most recent from every publications and composing tournaments out there. And everything in between (Troitsky, Rinck, Platov brothers, Kubbel, Kasparyan, etc..).
Back when I bought it in 2004 it contained 58000 chess endgame studies, but the most recent versions have most likely over 100k.
The lichess puzzles are free to download and use, with FEN and annotation of themes, openings, ratings, ... However, they do not come as a pgn file, but as a csv file, so you need to process the file first if you intend to open it using a regular PGN viewing software.