In this work, I provide a literature review of earlier works on classification of topological phases. In doing so, we motivate the idea of topological order and phases of matter that are dependent on global geometric features of systems in consideration. I review an algebraic formalism to categorize phases of matter containing non-abelian anyons, known as Braided Tensor Categories (BTC). With the motivation to study the relationship between non-abelian anyonic and symmetry defects, I also introduce the an algebraic formalism describing the interplay of global symmetries with the anyonic degrees of freedom. This formalism is known as $G$-crossed BTC. These algebraic formalism provide observable topological quantities, such as the topological $S$-matrix, that can be probed by interference measurements. When the topological $S$-matrix of the BTC is unitary, it is said that the fusion theory is said to be modular. Hence, one calls the algebraic formalism a Modular Tensor Category (MTC). I present the results of our work showing that one can construct projective representations of the mapping class group of the punctured torus from the topological formalism given by $G$-crossed BTC. This is pivotal in constructing a $G$-crossed BTC for surfaces of higher genus, as one can build surfaces of arbitrary genus by gluing many punctured tori. Throughout my mathematical analysis, I also show that modularity of BTCs are not destroyed by the presence of symmetry defects.