  A graph is said to be {\em vertex-transitive non-Cayley} if its full
automorphism group acts transitively on its vertices and contains no subgroups
acting regularly on its vertices. In this paper, a complete classification of
cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime,
is given. As a result, there are $11$ sporadic and one infinite family of such
graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the
infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family
there is a unique graph for a given order.
