  These expository notes, addressed to non-experts, are intended to present
some of Hironaka's ideas on his theorem of resolution of singularities. We
focus particularly on those aspects which have played a central role in the
constructive proof of this theorem.
  In fact, algorithmic proofs of the theorem of resolution grow, to a large
extend, from the so called Hironaka's fundamental invariant. Here we underline
the influence of this invariant in the proofs of the natural properties of
constructive resolution, such as: equivariance, compatibility with open
restrictions, with pull-backs by smooth morphisms, with changes of the base
field, independence of the embedding, etc.
