  For $p\in (1,\infty)$, we study representations of \'etale groupoids on
$L^{p}$-spaces. Our main result is a generalization of Renault's disintegration
theorem for representations of \'etale groupoids on Hilbert spaces. We
establish a correspondence between $L^{p}$-representations of an \'etale
groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight
regular $L^{p}$-representations of any countable inverse semigroup of open
slices of $G$ that is a basis for the topology of $G$. We define analogs
$F^{p}(G)$ and $F_{\mathrm{red}}^{p}(G)$ of the full and reduced groupoid
C*-algebras using representations on $L^{p}$-spaces. As a consequence of our
main result, we deduce that every contractive representation of $F^{p}(G)$ or
$F_{\mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of
our construction include the following natural families of Banach algebras:
discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on
$L^{p}$-spaces, and the analogs of AF-algebras on $L^{p}$-spaces. Our results
yield new information about these objects: their matricially normed structure
is uniquely determined. More generally, groupoid $L^{p}$-operator algebras
provide analogs of several families of classical C*-algebras, such as
Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras.
