  We present three equivalent definitions of $S^1$-equivariant symplectic
homology. We show that, using rational coefficients, the positive part of
$S^1$-equivariant symplectic homology is isomorphic to linearized contact
homology, when the latter is defined. We present several computations and
applications, and introduce a rigorously defined substitute for
cylindrical/linearized contact homology based on an $S^1$-equivariant
construction.
