The dynamics of relativistic stars and black holes are often studied in terms
of the quasinormal modes (QNM's) of the Klein-Gordon (KG) equation with
different effective potentials V(x). In this paper we present a systematic
study of the relation between the structure of the QNM's of the KG equation and
the form of V(x). In particular, we determine the requirements on V(x) in
order for the QNM's to form complete sets, and discuss in what sense they form
complete sets. Among other implications, this study opens up the possibility of
using QNM expansions to analyse the behavior of waves in relativistic systems,
even for systems whose QNM's do {\it not} form a complete set. For such
systems, we show that a complete set of QNM's can often be obtained by
introducing an infinitesimal change in the effective potential
