  We give necessary and sufficient conditions for the existence of a BMO
solution to the quasilinear equation $-\Delta_{p} u = \mu$ in $\mathbb{R}^n$,
$u\ge 0$, where $\mu$ is a locally finite Radon measure, and $\Delta_{p}u=
\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$).
  We also characterize BMO solutions to equations $-\Delta_{p} u = \sigma u^{q}
+ \mu$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $\mu$ and $\sigma$
are locally finite Radon measures. Our main results hold for a class of more
general quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ in
place of $\Delta_{p}$.
