  In this survey paper, we discuss the classical Cassels' theorem on existence
of small-height zeros of quadratic forms over Q and its many extensions, to
different fields and rings, as well as to more general situations, such as
existence of totally isotropic small-height subspaces. We also discuss related
recent results on effective structural theorems for quadratic spaces, as well
as Cassels'-type theorems for small-height zeros of quadratic forms with
additional conditions. We conclude with a selection of open problems.
