This project aims to study graphs commonly referred to as algebraically defined graphs. Many different families of these graphs have been shown to have remarkable properties, but many open questions remain. This project aims to answer questions in three areas: (A) Spectral Graph Theory, (B) Extremal Graph Theory and (C) Finite Geometry. (A) The family of algebraically defined graphs, denoted by D(k, q), has been conjectured to be nearly Ramanujan for all positive integers k. The ultimate goal of this project in this area is to determine the validity of the conjecture. There is much evidence that the conjecture is correct, and it has been verified for k = 2, 3, 4, 5. The most recent case was solved by my fellow graduate student and I. (B) Algebraically defined graphs are well known for their application to constructing dense graphs not containing some given graph as a subgraph. This project aims to improve known lower bounds for graphs without theta graphs of a particular size and for graphs not containing an 8-cycle as a subgraph. (C) In finite geometry, the key point focus of this project will be to determine the existence of particular projective planes and generalized quadrangles through the lens of algebraically defined graphs. This project will allow me to use my current expertise as well as to diversify my skill set which will allow me to be a versatile researcher with expertise in multiple areas.