In this paper we characterize manifolds (topological or smooth,
compact or not, with or without boundary) which admit flows having
a dense orbit (such manifolds and flows are called transitive)
thus fully answering some questions by Smith and Thomas. Namely,
it is shown that a surface admits a transitive flow (which can be
got smooth) if and only if it is connected and it is neither
homeomorphic to the sphere nor the projective plane nor embeddable
in the Klein bottle (or, alternatively, if it is connected and
includes two orientable topological circles intersecting
transversally at exactly one point). We also prove that any
(connected) manifold with dimension at least 3 admits a transitive
flow, which can be got smooth if the manifold admits a smooth
structure.
In particular, this allows us to characterize $\omega$-limit sets
with nonempty interior for flows in a given $n$-manifold (as
they can be described by the property of being the closure of
its transitive $n$-submanifolds).