  Let $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We
prove log-free zero density estimates for Rankin-Selberg $L$-functions of the
form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given family and $\pi_0$ is
fixed. These estimates are unconditional in many cases of interest; they hold
in full generality assuming an average form of the generalized Ramanujan
conjecture. We consider applications of these estimates related to mass
equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of
Rankin-Selberg $L$-functions, the Chebotarev density theorem, and
$\ell$-torsion in class groups of number fields.
