In this dissertation three mathematical studies on the evolution of human immunodeficiency virus (HIV) infection are presented. We use a variety of mathematical approaches (agent-based models, differential equation models, and a hybrid deterministic-stochastic algorithm) to provide insight into the biological dynamics of infection and virus evolution. Multiple infection (where cells can be infected with multiple copies of virus) has been documented to occur in HIV infection both \textit{in vitro} and \textit{in vivo} from human tissue samples. When a cell is infected with two or more viruses that are genetically different, then the process of recombination can occur, which has the potential to bring separate point mutations together in a single virus genome that previously were present in different genomes. It has been shown that both multiple infection and recombination can be promoted by direct cell-to-cell transmission of the virus through virological synapses (known as synaptic transmission), as synaptic transmission typically involves the simultaneous transfer of multiple viruses from the source cell to the target cell. Multiple infection, recombination, and synaptic transmission are therefore potentially important contributors to virus evolution, which can lead to the infection escaping from immune system responses, forming drug resistance, and overcoming potential vaccines. \ In Chapter 1, we demonstrate two important effects of the mode of viral spread: (i) For disadvantageous mutants, synaptic transmission protects against detrimental effects of recombination on double mutant persistence. Under free virus transmission, recombination increases double mutant levels for negative epistasis, but reduces them for positive epistasis. (ii) The mode of virus spread also directly influences the evolutionary fate of double mutants. For disadvantageous mutants, double mutant production is the predominant driving force, and hence synaptic transmission leads to highest double mutant levels due to increased transmission efficiency. For advantageous mutants, double mutant spread is the most important force, and hence free virus transmission leads to fastest invasion due to better mixing. For neutral mutants, both production and spread of double mutants are important, and hence an optimal mixture of free virus and synaptic transmission maximizes double mutant fractions. \ In Chapter 2, we use experimental data and parameterized mathematical models to show that cell-to-cell transmission operates by (i) increasing infection multiplicity, (ii) promoting the co-transmission of different virus strains from cell to cell, and (iii) increasing the rate at which point mutations are generated as a result of more reverse transcription events. This work further resulted in the estimation of various parameters that characterize important evolutionary processes, for example, we estimate that during cell-to-cell transmission an average of 3 viruses successfully integrated into the target cell, which can significantly raise the infection multiplicity compared to free virus transmission. \ In Chapter 3, we describe and implement a hybrid stochastic-deterministic algorithm to stochastically simulate mutant evolution in large viral populations. We demonstrate that the hybrid method can approximate the fully stochastic dynamics with sufficient accuracy at a fraction of the computational time, and quantify evolutionary end points that cannot be expressed by deterministic models. We apply this method to study the role of multiple infection and intracellular interactions among different virus strains (such as complementation and interference) for mutant evolution. Multiple infection is predicted to increase the number of mutants at a given infected cell population size, due to a larger number of infection events. We further find that viral complementation can significantly enhance the spread of disadvantageous mutants, but only in select circumstances: it requires the occurrence of direct cell-to-cell transmission through virological synapses, as well as a substantial fitness disadvantage of the mutant, most likely corresponding to defective virus particles.