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In the last decade, it was realized that various interacting conformal field theories saturate bounds on the CFT data coming from the numerical conformal bootstrap. In that light, finding an analytic explanation of the bootstrap bounds would give us an analytic handle on the physics of strongly-coupled CFTs. I will describe an approach to the conformal bootstrap based on the so-called analytic extremal functionals. This approach bridges the gap between the numerical and analytical conformal bootstrap and in particular provides an analytical explanation for various optimal bootstrap bounds. I will use the method to derive nonperturbative results concerning the distribution of primary operators in general CFTs and compute Witten diagrams in AdS_2 up to two loops. I will also discuss how this method provides a derivation of the conformal bootstrap based on Witten exchange diagrams.

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