Spaces of CFTs and conformal bootstrap

Conformal field theories can be seen as well-defined mathematical objects constrained by the bootstrap axioms. An interesting case is families of CFTs interpolating between perturbative and non-perturbative regimes. I will first review the case of the Ising CFT, where the conformal bootstrap has led to a comprehensive picture of the theory across dimensions between 2d and d=4-epsilon. I will then present new results for the tricritical Ising CFT, which is the IR fixed-point of phi^6 theory and interpolates between a 3-epsilon expansion and the tricritical minimal model in 2d.