We introduce a new class of quasilinear nonlocal operators and study
equations involving these operators. The operators are degenerate elliptic and
may have arbitrary growth in the gradient. Included are new nonlocal versions
of p-Laplace, ∞-Laplace, mean curvature of graph, and even strongly
degenerate operators, in addition to some nonlocal quasilinear operators
appearing in the existing literature. Our main results are comparison,
uniqueness, and existence results for viscosity solutions of linear and fully
nonlinear equations involving these operators. Because of the structure of our
operators, especially the existence proof is highly non-trivial and
non-standard. We also identify the conditions under which the nonlocal
operators converge to local quasilinear operators, and show that the solutions
of the corresponding nonlocal equations converge to the solutions of the local
limit equations. Finally, we give a (formal) stochastic representation formula
for the solutions and provide many examples
