  A vertex $v\in V$ is said to resolve two vertices $x$ and $y$ if $d_G(v,x)\ne
d_G(v,y)$. A set $S\subset V$ is said to be a metric generator for $G$ if any
pair of vertices of $G$ is resolved by some element of $S$. A minimum metric
generator is called a metric basis, and its cardinality, $\dim(G)$, the
\emph{metric dimension} of $G$. A set $S\subseteq V$ is said to be a
simultaneous metric generator for a graph family ${\cal
G}=\{G_1,G_2,\ldots,G_k\}$, defined on a common (labeled) vertex set, if it is
a metric generator for every graph of the family. A minimum cardinality
simultaneous metric generator is called a simultaneous metric basis, and its
cardinality the simultaneous metric dimension of ${\cal G}$. We obtain sharp
bounds for this invariants for general families of graphs and calculate closed
formulae or tight bounds for the simultaneous metric dimension of several
specific graph families. For a given graph $G$ we describe a process for
obtaining a lower bound on the maximum number of graphs in a family containing
$G$ that has simultaneous metric dimension equal to $\dim(G)$. It is shown that
the problem of finding the simultaneous metric dimension of families of trees
is $NP$-hard. Sharp upper bounds for the simultaneous metric dimension of trees
are established. The problem of finding this invariant for families of trees
that can be obtained from an initial tree by a sequence of successive
edge-exchanges is considered. For such families of trees sharp upper and lower
bounds for the simultaneous metric dimension are established.
