The purpose of the present paper is to contribute to the sequential theory of testing hypotheses about stochastic processes with a continuous parameter (say, $t$ which one may think of as time). Sequential decision problems about such processes seem not to have been treated before. Subsequently we shall treat problems of point and interval estimation and general sequential decision problems for such processes. The results, in addition to their interest per se and their practical importance, also shed light on the corresponding results for discrete stochastic processes. The subjects of sequential analysis and the theory of decision functions were founded by Wald, and we treat our present subjects in the spirit of his approach. The general results of decision theory, such as the complete class theorem, carry over to sequential problems about stochastic processes with continuous time parameter. As specific examples we treat the Wiener and Poisson processes and obtain, for example, the exact power function. (For discrete processes the corresponding known results, due to Wald, are approximations).