  The boundedness and compactness of weighted composition operators on the
Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference
is made to the case when the self-map of the disc is an inner function.
Schatten-class membership is also considered; as a result, stronger forms of
the two main results of a recent paper of Gunatillake are derived. Finally,
weighted composition operators on weighted Bergman spaces $\mathcal{A}^2
\alpha(\mathbb{D})$ are considered, and the results of Harper and Smith,
linking their properties to those of Carleson embeddings, are extended to this
situation.
