We present an approach to the general, non-Markovian dynamic allocation (or multiarmed bandit) problem, formulated in continuous time as a problem of stochastic control for multiparameter processes in the manner of Mandelbaum. This approach is based on a direct, martingale study of auxiliary questions in optimal stopping. Using a methodology similar to that of Whittle and relying on simple time-change arguments, we construct Gittins-index-type strategies, verify their optimality, provide explicit expressions for the values of dynamic allocation and associated optimal stopping problems, explore interesting dualities and derive various characterizations of Gittins indices. This paper extends results of our recent work on discrete-parameter dynamic allocation to the continuous time setup; it can be read independently of that work.