Inspect the full circumference and both sides of each tyre. Look for cuts, tears and sidewall bulges. Damage that allows the inner tube to protrude calls for tyre replacement. The rear tyre usually wears faster, so assess the two wheels separately.
If casing threads or the puncture-protection layer are showing through the rubber, replace the tyre. There is no single reliable mileage limit: load, surface, temperature, pressure and riding style all affect wear. Mountain-bike tread also matters for grip off road.
Some road tyres have small recessed wear indicators. When these wear away, replacement is due. Check the instructions for the particular model; distinguish an indicator from an accidental cut, and do not wait for it to disappear if the tyre is already damaged.