  In this paper, we establish various maximum principles and develop the method
of moving planes and the sliding method (on general unbounded domains) for
equations involving the uniformly elliptic nonlocal Bellman operator. As a
consequence, we derive multiple applications of these maximum principles and
the moving planes method. For instance, we prove symmetry, monotonicity and
uniqueness results and asymptotic properties for solutions to various equations
involving the uniformly elliptic nonlocal Bellman operator in bounded domains,
unbounded domains, epigraph or $\mathbb{R}^{n}$. In particular, the uniformly
elliptic nonlocal Monge-Amp\`{e}re operator introduced by Caffarelli and Charro
in \cite{CC} is a typical example of the uniformly elliptic nonlocal Bellman
operator.
