We propose a system, the Continuator, that bridges the gap between two classes of traditionally incompatible musical systems: (1) interactive musical systems, limited in their ability to generate stylistically consistent material, and (2) music imitation systems, which are fundamentally not interactive. Our purpose is to allow musicians to extend their technical ability with stylistically consistent, automatically learnt material. This goal requires the ability for the system to build operational representations of musical styles in a real time context. Our approach is based on a Markov model of musical styles augmented to account for musical issues such as management of rhythm, beat, harmony, and imprecision. The resulting system is able to learn and generate music in any style, either in standalone mode, as continuations of musician's input, or as interactive improvisation back up. Lastly, the very design of the system makes possible new modes of musical collaborative playing. We describe the architecture, implementation issues and experimentations conducted with the system in several real world contexts.