  We study the initial value problem for the integrable nonlocal nonlinear
Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2\sigma
q^{2}(x,t)\bar{q}(-x,t)=0 \] with decaying (as $x\to\pm\infty$) boundary
conditions. The main aim is to describe the long-time behavior of the solution
of this problem. To do this, we adapt the nonlinear steepest-decent method
\cite{DZ} to the study of the Riemann-Hilbert problem associated with the NNLS
equation. Our main result is that, in contrast to the local NLS equation, where
the main asymptotic term (in the solitonless case) decays to $0$ as
$O(t^{-1/2})$ along any ray $x/t=const$, the power decay rate in the case of
the NNLS depends, in general, on $x/t$, and can be expressed in terms of the
spectral functions associated with the initial data.
