  The main results presented in this dissertation are the following
  - We have shown that in $d=4$ weak hyperkahler torsion structures are the
same that hypercomplex structures and the same that the Plebanski-Finley
conformally invariant heavens. With the help of this identification we have
found the most general local form of an hyperkahler torsion space in four
dimensions. We also presented an Ashtekar like formulation for them in which to
finding an hyperkahler torsion metric is reduced to solve a quadratic
differential system.
  - It is found the most general form for the target space metric to the moduli
space metric of several $(n>1)$ identical matter hypermultiplets for the
type-IIA superstrings compactified on a Calabi-Yau threefold, near conifold
singularities, even taking into account non-perturbative D-instanton quantum
corrections. The metric in consideration is "toric hyperkahler" if we do not
take into account the gravitational corrections.
  - It is constructed a family of toric hyperkahler spaces in eight dimensions.
This spaces are lifted to backgrounds of the eleven dimensional supergravity
preserving certain ammount of supersymmetry, with and without the presence of
fluxes. Several type IIA and IIB backgrounds have been found by reduction along
a circle and by use of T-duality rules.
  - It is constructed a family of $Spin(7)$, $G_2$ and weak $G_2$ holonomy
metrics. The result has been lifted to a supergravity solution preserving one
supersymmetry. The presence of a toric symmetry allows a reduction to type IIA
backgrounds by usual reduction along one of the Killing vectors.
