  We consider the first Robin eigenvalue $\l_p(M,\a)$ for the $p$-Laplacian on
a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in
\R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems
of Cheng type for $\l_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower
bound of $\l_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower
bound and boundary mean curvature lower bound, via comparison with an
associated one-dimensional eigenvalue problem. The lower bound becomes an upper
bound when $\a<0$. Our results cover corresponding comparison theorems for the
first Dirichlet eigenvalue of the $p$-Laplacian when letting $\a \to +\infty$.
