Complex transition metal compounds (TMCs) research has produced functional materials with a range of properties, including ferroelectricity, colossal magnetoresistance, nonlinear optical activity and high-temperature superconductivity. Conventional routes to tune properties in transition metal oxides, for example, have relied primarily on cation chemical substitution and interfacial effects in thin film heterostructures. In heteroanionic TMCs, exhibiting two chemically distinct anions coordinating the same or different cations, engineering of the anion sub-lattice for property control is a promising alternative approach. The presence of multiple anions provides additional design variables, such as anion order, that are absent in homoanionic counterparts. The more complex structural and chemical phase space of heteroanionic materials provides a unique opportunity to realize enhanced or unanticipated electronic, optical, and magnetic responses. Although there is growing interest in heteroanionic materials, and synthetic and characterization advances are occurring for these materials, the crystal-chemistry principles for realizing structural and property control are only slowing emerging. This dissertation employs anion engineering to investigate phenomena in transition metal fluorides and oxyfluorides compounds using first principles density functional theory calculations. Oxyfluorides are particularly intriguing owing their tendency to stabilize highly ordered anion sublattices as well as the potential to combine the advantageous properties of transition metal oxides and fluorides. This work 1) addresses the challenges of studying fluorides and oxyfluorides using first principles calculations; 2) evaluates the feasibility of using external stimuli, such as epitaxial strain and hydrostatic pressure, to control properties of fluorides and oxyfluorides; and 3) formulates a computational workflow based on multiple levels of theory and computation to elucidate structure-property relationships and anion-order descriptors. The insights gained in this work advance the understanding of oxide-fluoride anion engineered materials and we anticipate that it will motivate novel experimental efforts and materials by design in the future.