  This thesis is dedicated to the study of the geometry of six-dimensional
superspace, endowed with the minimal amount of supersymmetry. In the first part
of it, we unfold the main geometrical features of such superspace by solving
completely the Bianchi identities for the constrained superspace torsion, which
allow us to determine the full six-dimensional derivate superalgebra. Next, the
conformal structure of the supergeometry is considered. Specifically, it is
shown that the conventional torsion constraints remain invariant under
super-Weyl transformations generated by a real scalar superfield parameter. In
the second part of this work, the field content and superconformal matter
couplings of the supergeometry are explored. The component field content of the
Weyl multiplet is presented and the question of how this multiplet emerges in
superspace is addressed. Finally, the constraints that conformal invariance
imposes on some matter representations are analyzed.
