  We present a new criterion for the complex hyperbolicity of a non-compact
quotient X of a bounded symmetric domain. For each p $\ge$ 1, this criterion
gives a precise condition under which the subvarieties V $\subset$ X with dim V
$\ge$ p are of general type, and X is p-measure hyperbolic. Then, we give
several applications related to ball quotients, or to the Siegel moduli space
of principally polarized abelian varieties. For example, we determine effective
levels l for which the moduli spaces of genus g curves with l-level structures
are of general type.
