Run-time adaptability is a key feature of dynamic business environments, where the processes need to be constantly refined and restructured to deal with context changes. In this paper, we present CAptLang, a language to model context-aware and adaptable business processes where the main feature is the possibility of leaving the handling of extraordinary or improbable situations to run time. We present CAptLang with its formal syntax and semantics. Moreover we show how its semantics have been used to guide the implementation of a Java-based business processes execution engine, component of the ASTRO-CAptEvo adaptation framework.