  Hironaka's resolution of singularities is a celebrated discipline that has
many applications in many areas of Mathematics. In this work we approach the
desingularization of semialgebraic sets ${\mathcal S}\subset{\mathbb R}^n$. We
prove the counterpart to `Hironaka's desingularization of real algebraic sets'
in the semialgebraic setting. It involves Nash manifolds with corners instead
of non-singular real algebraic sets. The obtained results are optimal when
${\mathcal S}$ is a closed semialgebraic set. The main tools involved are:
Hironaka's desingularization of real algebraic sets, Hironaka's embedded
desingularization of real algebraic subsets of non-singular real algebraic sets
and drilling blow-ups of Nash manifolds with centers closed Nash submanifolds.
  We show how to `build' a Nash manifold with corners ${\mathcal
Q}\subset{\mathbb R}^n$ from a suitable Nash manifold $M\subset{\mathbb R}^n$
(of its same dimension), which contains ${\mathcal Q}$ as a closed subset, by
folding $M$ along the irreducible components of a normal-crossings divisor of
$M$, which is the smallest Nash subset of $M$ that contains the boundary
$\partial{\mathcal Q}$ of ${\mathcal Q}$. One can choose as the Nash manifold
$M$ the Nash `double' $D({\mathcal Q})$ of ${\mathcal Q}$, which is analogous
to the Nash double of a Nash manifold with boundary, but $D({\mathcal Q})$
takes into account the peculiarities of the boundary of a Nash manifold with
corners.
  We include several applications: (1) Weak desingularization of closed
semialgebraic sets using Nash manifolds with boundary, (2) Representation of
compact semialgebraic sets connected by analytic paths as images under Nash
maps of closed unit balls, (3) Construction of Nash models for compact
orientable smooth surfaces of genus $g\geq0$, and (4) Nash approximation of
continuous semialgebraic maps whose target spaces are Nash manifolds with
corners.
