A separable and metrizable space X is called a flowbox manifold if there exists a base for the open sets each of whose elements has a product structure with the reals ℜ as a factor such that a natural consistency condition is met. We show how flowbox manifolds can be divided into orientable and nonorientable ones. We prove that a space X is an orientable flowbox manifold if and only if X can be endowed with the structure of a flow without restpoints. In this way we generalize Whitney’s theory of regular families of curves so as to include self-entwined curves in general separable metric spaces. All spaces under consideration are separable and metrizable. © 1991 American Mathematical Society.