  Given a connected simple graph $G=(V,E)$, and a positive integer $k$, a set
$S\subseteq V$ is said to be a $k$-metric generator for $G$ if and only if for
any pair of different vertices $u,v\in V$, there exist at least $k$ vertices
$w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, for every $i\in
\{1,...,k\}$, where $d_G(x,y)$ is the length of a shortest path between $x$ and
$y$. A $k$-metric generator of minimum cardinality in $G$ is called a
$k$-metric basis and its cardinality, the $k$-metric dimension of $G$. In this
article we study the $k$-metric dimension of corona product graphs
$G\odot\mathcal{H}$, where $G$ is a graph of order $n$ and $\mathcal{H}$ is a
family of $n$ non-trivial graphs. Specifically, we give some necessary and
sufficient conditions for the existence of a $k$-metric basis in a connected
corona graph. Moreover, we obtain tight bounds and closed formulae for the
$k$-metric dimension of connected corona graphs.
