  A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint
copies of $K_r$ that together cover all the vertices in $G$. In this paper we
consider perfect $K_r$-tilings in the setting of randomly perturbed graphs; a
model introduced by Bohman, Frieze and Martin where one starts with a dense
graph and then adds $m$ random edges to it. Specifically, given any fixed $0<
\alpha <1-1/r$ we determine how many random edges one must add to an $n$-vertex
graph $G$ of minimum degree $\delta (G) \geq \alpha n$ to ensure that,
asymptotically almost surely, the resulting graph contains a perfect
$K_r$-tiling. As one increases $\alpha$ we demonstrate that the number of
random edges required `jumps' at regular intervals, and within these intervals
our result is best-possible. This work therefore closes the gap between the
seminal work of Johansson, Kahn and Vu (which resolves the purely random case,
i.e., $\alpha =0$) and that of Hajnal and Szemer\'edi (which demonstrates that
for $\alpha \geq 1-1/r$ the initial graph already houses the desired perfect
$K_r$-tiling).
