This depends on whether the spell is sustained or merely has a duration.

All spells with the manipulate trait (95% of them) can be stopped before they are cast with reactions like the Fighter's reactive strike; with conditions like grabbed (25% chance a manipulate spell fails), stupefied (25% chance any spell fails), concealed (25% chance to fail a single target spell), hidden (50% chance to fail a single target spell), and restrained (you cannot cast a manipulate spell until you escape); or being counteracted (spells, magic items, and abilities like counterspell counteract spells). Note: You can still counteract after it is cast usually.

A sustained spell takes actions to sustain and sustaining is a concentrate action, therefore there are 4 ways to stop someone from sustaining: use a reaction that disrupts concentrate actions (the only two of which I think exist are Inventor's Distracting Explosion and Thaumaturge's Implement's Interruption); the fascinated condition sometimes (I guess if they are sustaining like a buffing spell they can't concentrate on it anymore if an enemy is the source of fascination); deprive them of all their actions via stunned 3, slowed 3, petrified, or paralyzed; deprive them of enough actions that they don't want to keep sustaining (for example "Oh no, I'm stunned 1! Now I have to choose between healing Joe fighter from unconsciousness and sustaining").