  We introduce a notion of a girth-regular graph as a $k$-regular graph for
which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called
the signature) giving, for every vertex $u$ of the graph, the number of girth
cycles the edges with end-vertex $u$ lie on. Girth-regularity generalises two
very different aspects of symmetry in graph theory: that of vertex transitivity
and that of distance-regularity. For general girth-regular graphs, we give some
results on the extremal cases of signatures. We then focus on the cubic case
and provide a characterisation of cubic girth-regular graphs of girth up to
$5$.
