  We study the heat trace asymptotics associated with the Steklov eigenvalue
problem on a Riemannian manifold with boundary. In particular, we describe the
structure of the Steklov heat invariants and compute the first few of them
explicitly in terms of the scalar and mean curvatures. This is done by applying
the Seeley calculus to the Dirichlet-to-Neumann operator, whose spectrum
coincides with the Steklov eigenvalues. As an application, it is proved that a
three--dimensional ball is uniquely defined by its Steklov spectrum among all
Euclidean domains with smooth connected boundary.
