  For a graph $F$, we say that another graph $G$ is $F$-saturated, if $G$ is
$F$-free and adding any edge to $G$ would create a copy of $F$. We study for a
given graph $F$ and integer $n$ whether there exists a regular $n$-vertex
$F$-saturated graph, and if it does, what is the smallest number of edges of
such a graph. We mainly focus on the case when $F$ is a complete graph and
prove for example that there exists a $K_3$-saturated regular graph on $n$
vertices for every large enough $n$.
  We also study two relaxed versions of the problem: when we only require that
no regular $F$-free supergraph of $G$ should exist or when we drop the $F$-free
condition and only require that any newly added edge should create a new copy
of $F$.
