The defence that a person who knowingly took on a danger cannot complain of it - and the signed waiver that tries to make that agreement explicit before the gym, the ski slope or the trampoline park.
Assumption of risk is a defence to an injury claim. In its "express" form it is a contract: the injured person signed a waiver or release agreeing in advance not to hold the operator liable for injuries from the activity. In its "implied" form it is inferred from conduct: a person who voluntarily took part in an activity whose inherent dangers were obvious - being struck by a foul ball at a baseball game, falling while skiing - is treated as having accepted those inherent risks, and the operator owes no duty to protect against them.
The distinction the law draws is between the risks inherent in the activity and risks the operator added by carelessness. A ski resort is generally not liable for a fall on a properly maintained run; it may well be liable for an unmarked snow-making pipe across it. Most states now treat implied assumption of risk, where it is not a complete "no duty" bar, as a form of the injured person's own fault, to be weighed under the comparative-negligence rule rather than barring the claim outright.
Signed waivers are enforced in most states if they are clear, conspicuous and cover the injury that occurred, but every state has limits. Nearly all refuse to enforce a waiver of liability for gross negligence, recklessness or intentional harm; several will not enforce a waiver signed by a parent on behalf of a child; a few refuse to enforce pre-injury releases altogether as against public policy; and ambiguous wording is read against the operator who drafted it. Whether a waiver covers the defendant's own negligence at all is, in many states, a question of whether it says so plainly.
A signed waiver is not the end of the question, and an operator who says otherwise is stating its position, not the law. A lawyer reads the document against the state's rules on scope, conspicuousness and gross negligence, and looks at whether the injury came from an inherent risk or from something the operator did or failed to do - which is usually where the case is.
Choose your state. Each link opens the directory page for the city in that state with the most currently published law firms in this practice area; a +n beside the city is how many other cities in the state also have one. The list is generated when this page loads, so a state whose listings have lapsed drops out rather than becoming a dead link.