Exceptional vector bundles are a class of rigid bundles commonly found on Fano varieties with close connections to geometry and homological algebra. Foundational results of Drezet and Le Potier from the 1980s show that on the projective plane exceptional bundles have a deep and elaborate relationship with stable bundles in general. In this talk I will explore how these results in some ways extend and in other ways fail to extend to three-dimensional projective space, and discuss several interesting open problems concerning exceptional bundles.