In this thesis, we explore analytical methods to study conformal field theories (CFTs) in a general number of spacetime dimensions. We first use the lightcone bootstrap to systematically study correlation functions of scalar operators charged under global symmetries. We then generalize existing techniques in the lightcone bootstrap to study four-point functions containing operators with spin. As an application, we observe a close connection between anomalous dimensions of large spin, double-twist operators and the conformal collider bounds. Through further refinement of these techniques and the application of known analyticity properties for four-point functions, we also present a proof for these bounds that relies on basic physical consistency conditions. We then generalize these techniques further to study large N CFTs in the Regge limit and the implications of crossing symmetry in this limit. By studying the Regge limit, we can make new predictions for the large twist, large spin spectrum of CFTs and derive new bounds on CFT data. In the final part of this thesis, we use the Regge limit and constraints from unitarity to derive new bounds for both large N and generic CFTs. For large N CFTs, we derive new constraints on theories dual to a weakly-coupled, gravitational theory in an Anti-deSitter (AdS) spacetime, and for generic CFTs we derive generalizations of the conformal collider bounds.
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