Mostly your eyes and your nose, with one catch: you need to know what normal looks like for that particular cheese. A washed-rind cheese is supposed to be a little sticky and smell like feet. A Brie is supposed to have a white mold rind. Neither is a warning sign.
Signs the cheese has turned:
- A slimy or sticky film on a cheese that isn't meant to have one (a firm cheddar, say).
- Mold in the wrong place or the wrong color. Fuzzy green, black or pink growth on a cheese that isn't mold-ripened. Whether you can cut it away depends on the cheese, so see Can you cut the mold off cheese?
- A sour, rancid or chemical smell that goes beyond the cheese's usual funk. (Ammonia on a Brie is its own story and often fixable.)
- An off, bitter taste. If one small taste is wrong, stop there.
Things that look alarming but usually aren't: a dry, hard or cracked surface (stale, but fine for cooking), little white crystals in an aged cheese (those are flavor crystals, a good sign), and beads of oil on the surface (the cheese got warm).
The simple rule: the softer and fresher the cheese, the less forgiving it is. If a soft or fresh cheese looks or smells wrong, toss it.
Related: How should I store leftover cheese? · Can cheese be frozen? · Should cheese be kept in the fridge? · Can you cut the mold off cheese? · How long does cheese last? · What are the crunchy white crystals in aged cheese?