  For $p \in [1, \infty),$ we define and study full and reduced crossed
products of algebras of operators on $\sigma$-finite $L^p$ spaces by isometric
actions of second countable locally compact groups. We give universal
properties for both crossed products. When the group is abelian, we prove the
existence of a dual action on the full and reduced $L^p$ operator crossed
products. When the group is discrete, we construct a conditional expectation to
the original algebra which is faithful in a suitable sense. For a free action
of a discrete group on a compact metric space $X,$ we identify all traces on
the reduced $L^p$ operator crossed product, and if the action is also minimal
we show that the reduced $L^p$ operator crossed product is simple. We prove
that the full and reduced $L^p$ operator crossed products of an amenable $L^p$
operator algebra by a discrete amenable group are again amenable. We prove a
Pimsner-Voiculescu exact sequence for the K-theory of reduced $L^p$ operator
crossed products by ${\mathbb{Z}}.$ We show that the $L^p$ analogs
${\mathcal{O}}_d^p$ of the Cuntz algebras ${\mathcal{O}}_d$ are stably
isomorphic to reduced $L^p$ operator crossed products of stabilized $L^p$ UHF
algebra by ${\mathbb{Z}},$ and show that $K_0 ({\mathcal{O}}_d^p) \cong
{\mathbb{Z}} / (d - 1) {\mathbb{Z}}$ and $K_1 ({\mathcal{O}}_d^p) = 0.$
