Very few laws specifically control police pursuits. Instead, police department policies dictate pursuit restrictions. Legally, police officers have a right to pursue. Even if they fail to follow department policy, they probably won't have broken any laws, but they might suffer consequences within the police force.
According to National Highway Traffic Safety Administration data, there were 365 deaths resulting from crashes during police chases in 2001. 140 of those killed were innocent bystanders [ref].
Who is liable if a chase results in property damage, injury or death? It depends on the laws of the town, city or county in which the chase occurred. The American common law rule is sovereign immunity -- that is, government officials and institutions are not liable for any damages that occur while they are in the course of their government duties, no matter how negligent their actions. However, most municipalities have written laws that supersede this sovereign immunity. These laws make individuals immune from liability, but claimants can still sue institutions. So if a police car injures your family during a chase, you can't sue the officer driving the car, but you can sue the police department and its municipality. Again, liability will vary from place to place. California is known for a law that gives police departments almost complete immunity in pursuit situations.
A common myth about police chases is that it is illegal for a police officer to drive above the speed limit and disobey traffic signs unless the lights and sirens are on. In fact, police officers can legally pursue at high speeds even if they don't turn on lights and sirens [ref]. However, department policies may contradict this precedent.
For lots more information about police chases, check out the links on the next page.