A classic result of Marstrand and Mattila shows that given a subset of n-dimensional Euclidean space, for almost every k-dimensional subspace, the size of the projection of the subset onto said subspace is large. Given this result, one might then ask: how often is the size of the projection smaller than expected? Answers to this question are known as exceptional set estimates. In this talk, we will begin by discussing some classical results due to Falconer/Peres-Schlag and Kaufman. We will give a proof of said classical results using Fourier analytic and double counting techniques due to B.-Gan '22. Similar exceptional set estimates can also be found for radial projections. As such, we will present radial projection conjectures put forward by Lund-Pham-Thu and Liu, which were later resolved by B.-Gan, Orponen-Shmerkin-Wang, and B.-Lund-Pham. We conclude with a discussion of current work and further directions regarding exceptional set estimates.