Saves are usually compatible within a given minor version; e.g. you can transfer a 0.34.08 save to 0.34.11 and it should work. There are sometimes glitches, though. For instance, if you tried to take a 0.34.07 save into 0.34.11 you would need to mess around in the raw files to add wheelbarrows and minecarts (or just not be able to make them at all, which would be rather bad for stone stockpiles).
Sometimes new features require generating a new world. Depends on the feature.
It's a safe bet that trying to transfer a save across versions (e.g. 0.31.x to 0.34.y) won't work.