The moduli space of string theories has been the subject of intense research efforts for many years now. Much of these efforts are focused on string compactifications, and in particular worldsheet conformal field theories (CFTs) embedded in toroidal target spaces. The CFTs defined at each point in this moduli space consist of an algebra of operators which define the content of the theory. In this thesis, we will investigate methods of traversing this moduli space, and we will attempt to elucidate the relations and symmetries that exist between different points in this space. Specifically, we study the deformation of operators in string compactifications, as well as T-duality from the worldsheet CFT perspective. We review, and bring into a contemporary context, a construction based on universal coordinates that can be used to define operators at one point in moduli space in terms of the operators at some reference point. We also review how this construction can be used to perform T-duality algebraically, thus providing an alternative perspective to the Buscher construction. Using the language of connections and parallel transport on the space of backgrounds, we discuss how to deform general operators in a given space of theories, including quantum field theories (QFTs) lacking conformal symmetry. We find that, for a general operator, there are two sources of deformation. The first is the usual deformation operator derived from the worldsheet sigma model. The second, less familiar part is a deformation directly induced as a result of the change in the background, which depends on the tensor structure of the operator of interest. In particular, scalar operators are invariant under this deformation. In the literature, since it is usually scalar operators such as the stress tensor that are of interest, this part of the deformation has not previously been addressed to our knowledge. Throughout, we apply our formalisms to well-known torus bundle examples such as the nilfold, the T3 with H-flux and the T-fold, and we also employ the doubled geometry construction. Initially, we utilise an ‘adiabatic’ approximation, where we neglect the worldsheet interactions arising from the coordinate dependence of the background. We investigate how the gauge algebra of the torus bundle with doubled fibres, pulled back to the worldsheet, compares with the algebra of the zero modes. Surprisingly, we find that the algebra of the worldsheet theory reproduces the doubled twisted torus algebra, i.e. the algebra where the base is also doubled. We also consider worldsheet sigma models corresponding to these torus bundles in their entirety, away from the adiabatic limit, and derive operator deformations in this context. We discuss T-duality between these backgrounds and we explain how our formalism could be used to construct T-dual backgrounds in more general settings. We also consider how the formalism applies to the N = 1 superstring in the NS-NS sector.