  A graph $G=(V,E)$ with geodesic distance $d(\cdot,\cdot)$ is said to be
resolved by a non-empty subset $R$ of its vertices when, for all vertices $u$
and $v$, if $d(u,r)=d(v,r)$ for each $r\in R$, then $u=v$. The metric dimension
of $G$ is the cardinality of its smallest resolving set. In this manuscript, we
present and investigate the notions of resolvability and metric dimension when
the geodesic distance is truncated with a certain threshold $k$; namely, we
measure distances in $G$ using the metric $d_k(u,v):=\min\{d(u,v),k+1\}$. We
denote the metric dimension of $G$ with respect to $d_k$ as $\beta_k(G)$. We
study the behavior of this quantity with respect to $k$ as well as the diameter
of $G$. We also characterize the truncated metric dimension of paths and cycles
as well as graphs with extreme metric dimension, including graphs of order $n$
such that $\beta_k(G)=n-2$ and $\beta_k(G)=n-1$. We conclude with a study of
various problems related to the truncated metric dimension of trees.
