  Let $X(\Delta)$ be the real toric variety associated to a smooth fan
$\Delta$. The main purpose of this article is: (i) to determine the fundamental
group and the universal cover of $X(\Delta)$, (ii) to give necessary and
sufficient conditions on $\Delta$ under which $\pi_1(X(\Delta))$ is abelian,
(iii) to give necessary and sufficient conditions on $\Delta$ under which
$X(\Delta)$ is aspherical, and when $\Delta$ is complete, (iv) to give
necessary and sufficient conditions for $\cc_{\Delta}$ to be a $K(\pi,1)$ space
where $\cc_{\Delta}$ is the complement of a real subspace arrangement
associated to $\Delta$.
