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We present techniques for incremental computing by introducing adaptive functional programming. As an {\em adaptive} program executes, the underlying system represents the data and control dependences in the execution in the form of a {\em dynamic dependence graph}. When the input to the program changes, a change propagation algorithm updates the output and the dynamic dependence graph by propagating changes through the graph and re-executing code where necessary. Adaptive programs adapt their output to any change in the input, small or large. We show that adaptivity techniques are practical by giving an efficient implementation as a small ML library. The library consists of three operations for making a program adaptive, plus two operations for making changes to the input and adapting the output to these changes. We give a general bound on the time it takes to adapt the output, and based on this, show that an adaptive Quicksort adapts its output in logarithmic time when its input is extended by one key. To show the safety and correctness of the mechanism we give a formal definition of \AFL{}, a call-by-value functional language extended with adaptivity primitives. The modal type system of \AFL{} enforces correct usage of the adaptivity mechanism, which can only be checked at run time in the ML library. Based on the \AFL{} dynamic semantics, we formalize the \ncpa{} and prove its correctness.