We consider a semilinear elliptic problem [- \Delta u + u = (I_\alpha \ast
\abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_\alpha)
is a Riesz potential and (p>1). This family of equations includes the Choquard
or nonlinear Schr\"odinger-Newton equation. For an optimal range of parameters
we prove the existence of a positive groundstate solution of the equation. We
also establish regularity and positivity of the groundstates and prove that all
positive groundstates are radially symmetric and monotone decaying about some
point. Finally, we derive the decay asymptotics at infinity of the
groundstates.Comment: 23 pages, updated bibliograph