KeepDoor Closed

Door closedimage

Else maybe use releasable adhesive to attach brackets that hold magnets to the door so that the magnets can hold the door shut.

KeepDoor ClosedSign

Not sure what your hinges look like but you can add a spring to a hinge to make it "auto close" (also not sure if this is strong enough for cats?)

This might not fit seamlessly into your situation, but there are spring loaded hinges that close a door. (The most common application here is for doors between garages and houses, as one doesn't want exhaust in a house.)

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My bathroom door doesn't have a handle, only a lock, and the door won't stay closed. As I have very curious cats, I can't just use a hook and keep it open, so right now, I've taped a piece of cardboard to the door so the pressure between the door and the frame keeps it closed.

KeepDoor ClosedSticker

Door ClosedSign

This is a very ugly solution and I want to improve it, but I don't own this apartment (and will probably leave it in 2 years), nor a drill, so I'd like to avoid solutions that need drilling.

If conditions are just right, even a draft stopper like the one pictured below might give enough friction for your purpose. (image credit: Amazon.com/VISEMAN)

Door closedclipart

This question has some interesting ideas, but most of them do use holes. I also thought about attaching magnets to the metallic lock part (pardon my English) and hope that even if it doesn't touch the frame, there will be enough force to resist the wind (and the cats). I don't really have high hopes though...

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You could replace the lock with a passage door knob set with or without a lock. Post a picture so we can get a better idea of what your problem is and maybe offer a better fix.

How about an object weighing a kilo or two? A rock.. a brick.. a book.. a bag of sugar, flour, beans, etc.. a container with a couple liters of water..

Attach a cord, ribbon, etc to the object so that when you're on the other side, where the door swings away from you, the cord can be passed through the gap below the door. Pull the cord while closing the door so that the weight is drawn up behind the door and holds it closed.