A quantum spin network can be modelled by an undirected graph whose vertices and edges represent qubits and their interactions in the network, respectively. One major problem involving quantum spin networks is determining a time   such that the state at one vertex is transported to another vertex at time   with a particular level of probability, called the fidelity of quantum state transfer. Various useful types of quantum state transfer arise depending on the fidelity: periodicity, perfect state transfer, pretty good state transfer, and fractional revival. In this thesis, we investigate quantum state transfer between twin vertices in weighted graphs with or without loops. We provide spectral and algebraic characterizations of twin vertices, and use these to identify which properties of twin vertices are essential for various types of quantum state transfer to occur between any two of them. A characterization of vertices in unweighted graphs that exhibit strong cospectrality with an additional vertex in the post-twinning graph is also given. Moreover, we determine necessary and sufficient conditions for periodicity, perfect state transfer, pretty good state transfer, and fractional revival to occur between twin vertices with respect to adjacency dynamics. Then, we apply our results to double cones on regular graphs, which are a special class of graphs with twin vertices. Finally, we explore quantum state transfer in some common families of graphs with respect to adjacency dynamics.