  The primary focus of this thesis is to investigate the mathematical and
physical properties of spaces that are related by T-duality and its
generalisations. In string theory, T-duality is a relationship between two a
priori different string backgrounds which nevertheless behave identically from
a physical point of view. These backgrounds can have different geometries,
different fluxes, and even be topologically distinct manifolds. T-duality is a
uniquely `stringy' phenomenon, since it does not occur in a theory of point
particles, and together with other dualities has been incredibly useful in
elucidating the nature of string theory and M-theory.
  There exist various generalisations of the usual T-duality, some of which are
still putative, and none of which are fully understood. Some of these dualities
are inspired by mathematics and some are inspired by physics. These
generalisations include non-abelian T-duality, Poisson-Lie T-duality,
non-isometric T-duality, and spherical T-duality. In this thesis we review
T-duality and its various generalisations, studying the geometric, topological,
and physical properties of spaces related by these dualities.
