This thesis explores the geometry of string theory backgrounds and the nongeometric features of string theory that arise due to T-duality. For this reason, it is divided into two complementary parts. Part I deals with the superalgebras of symmetries of string theory and M-theory backgrounds, the so-called Killing superalgebras. It is shown that one can define a Lie superalgebra consisting of the infinitesimal field-preserving isometries and the supersymmetries of the background. We also explore the extension of a Killing superalgebra with brane charges. Part II deals with non-geometric backgrounds. In particular, we adopt the framework of the doubled geometry, also known as the doubled torus. We analyze the hamiltonian dynamics of the system and quantize a model T-fold. Finally we extended the doubled torus system to include worldsheet supersymmetry. Throughout part II, we focus on the equivalence, classical and quantum, of the doubled formalism with the conventional formulation.