In the first part of this thesis, we investigate a way to find the complex structure moduli, for a given background of type IIB string theory in the presence of flux in special cases. We introduce a way to compute the complex structure and axion dilaton moduli explicitly. In the second part, we discuss $(0,2)$ supersymmetric versions of some recent exotic $mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models, describing intersections of Grassmannians. In the next part, we consider mirror symmetry for certain gauge theories with gauge groups $F_4$, $E_6$, and $E_7$. In the last part of this thesis, we study whether certain branched-double-cover constructions in Landau-Ginzburg models can be extended to higher covers. / Doctor of Philosophy / This thesis concerns string theory, a proposal for unification of general relativity and quantum field theory. In string theory, the building block of all the particles are strings, such that different vibrations of them generate particles. String theory predicts that spacetime is 10-dimensional. In string theorist's intuition, the extra six-dimensional internal space is so small that we haven't detected it yet. The physics that string theory predicts we should observe, is governed by the shape of this six-dimensional space called a `compactification manifold.' In particular, the possible ways in which this geometry can be deformed give rise to light degrees of freedom in the associated observable physical theory. In the first part of this thesis, we determine these degrees of freedom, called moduli, for a large class of solutions of the so-called type IIB string theory. In the second part, we focus on constructing such spaces explicitly. We also show that there can be different equivalent ways of constructing the same internal space. The third part of the thesis concerns mirror symmetry. Two compactification manifolds are called mirror to each other, when they both give the same four-dimensional effective theory. In this part, we describe the mirror of two-dimensional gauge theories with $F_4$, $E_6$, and $E_7$ gauge group, using the Gu-Sharpe proposal.