  In this work we study the existence of surjective Nash maps between two given
semialgebraic sets ${\mathcal S}$ and ${\mathcal T}$. Some key ingredients are:
the irreducible components ${\mathcal S}_i^*$ of ${\mathcal S}$ (and their
intersections), the analytic-path connected components ${\mathcal T}_j$ of
${\mathcal T}$ (and their intersections) and the relations between dimensions
of the semialgebraic sets ${\mathcal S}_i^*$ and ${\mathcal T}_j$. A first step
to approach the previous problem is the former characterization done by the
second author of the images of affine spaces under Nash maps.
  The core result of this article to obtain a criterion to decide about the
existence of surjective Nash maps between two semialgebraic sets is: {\em a
full characterization of the semialgebraic subsets ${\mathcal S}\subset{\mathbb
R}^n$ that are the image of the closed unit ball $\overline{\mathcal B}_m$ of
${\mathbb R}^m$ centered at the origin under a Nash map $f:{\mathbb
R}^m\to{\mathbb R}^n$}. The necessary and sufficient conditions that must
satisfy such a semialgebraic set ${\mathcal S}$ are: {\em it is compact,
connected by analytic paths and has dimension $d\leq m$}.
  Two remarkable consequences of the latter result are the following: (1) {\em
pure dimensional compact irreducible arc-symmetric semialgebraic sets of
dimension $d$ are Nash images of $\overline{\mathcal B}_d$}, and (2) {\em
compact semialgebraic sets of dimension $d$ are projections of non-singular
algebraic sets of dimension $d$ whose connected components are Nash
diffeomorphic to spheres (maybe of different dimensions)}.
