In many medical and health-care contexts, a failure event is triggered when a subject's deteriorating health first reaches a failure threshold. The failure process can be described mathematically as the sample path of a stochastic process hitting a boundary. The same kind of model describes failure events in other application fields, such as engineering. Threshold regression refers to first hitting time models that have built-in regression structures. To date, applications of threshold regression have been based on parametric families of stochastic processes. We will present a distribution-free form of threshold regression that requires the stochastic process to have only one key property, namely, stationary independent increments. As this property is frequently encountered in real applications, this threshold regression model and its related statistical methodology have potential for general application in many fields. Threshold regression in combination with Markov decomposition allows the distribution-free methods to be applied to longitudinal time-to-event data. Two examples will be presented.