The creation of protocols to form novel C-C and C-heteroatom bonds has been the primary goal of organic synthesis since its inception. Chemists have long harnessed both radical and polar reactivities, often as complementary paths to construct these bonds to yield more complex molecular architectures. However, compared to the development of synthetic protocols, development of mechanistic models and enriching of mechanistic understanding of many organic reactions has been limited. Computational studies into the mechanisms of organic transformations provide an avenue by which mechanisms of reactions can be better understood and new patterns of reactivity can be predicted. Herein, quantum-mechanical computational methods e.g., density functional theory (DFT) have been employed in the pursuit of understanding the mechanisms of a series of radical and polar reaction schemes. Specifically, DFT calculations were employed to understand the mechanism and origins of selectivity of two nickel(I)-catalyzed olefin functionalizations. These studies demonstrate a catalyst-control scheme by which selectivity can be induced by the steric properties of the catalyst (Chapter 1). Following this work, two photocatalytic transformations which yield difluorinated products were studied thoroughly with computations. First, a synthesis of difluorinated lactone derivatives revealed a long-lived radical intermediate and motivated mechanistic experiments to isolate this radical. Next, a synthesis of difluorinated oxindole derivatives demonstrated the ability of arenethiols to act as photocatalysts (Chapter 2). Then, computations were used to rigorously explore a copper-catalyzed reductive cross-coupling of imine and allenamides. Specifically, computations were employed to explore the mechanism of the transformation and the origins of stereoselectivity and the divergent formation of urea and diamine products (Chapter 3). Finally, two computational investigations into the mechanisms of transformations catalyzed by first-row transition metals are detailed. In particular, a nickel-catalyzed hydroarylation of gem-difluoroalkenes is explored computationally to determine the order of steps in the reaction. In addition, the mechanism of a cobalt(I)-catalyzed allylic substitution is considered to ascertain the nature of the transformation as either radical or polar (Chapter 4). Given the complexity of the mechanisms of these transformations, computational studies provide an alternative route to acquire useful mechanistic understanding that can support or explain observed experiments and suggest further mechanistic experiments that could provide stronger evidence for a given mechanistic proposal.