  We consider stochastic differential equations of the form
$dY_t=V(Y_t)\,dX_t+V_0(Y_t)\,dt$ driven by a multi-dimensional Gaussian
process. Under the assumption that the vector fields $V_0$ and
$V=(V_1,\ldots,V_d)$ satisfy H\"{o}rmander's bracket condition, we demonstrate
that $Y_t$ admits a smooth density for any $t\in(0,T]$, provided the driving
noise satisfies certain nondegeneracy assumptions. Our analysis relies on
relies on an interplay of rough path theory, Malliavin calculus and the theory
of Gaussian processes. Our result applies to a broad range of examples
including fractional Brownian motion with Hurst parameter $H>1/4$, the
Ornstein-Uhlenbeck process and the Brownian bridge returning after time $T$.
