  In this work we initiate the conformal bootstrap program for ${\mathcal N}=2$
superconformal field theories in four dimensions. We promote an abstract
operator-algebraic viewpoint in order to unify the description of Lagrangian
and non-Lagrangian theories, and formulate various conjectures concerning the
landscape of theories. We analyze in detail the four-point functions of flavor
symmetry current multiplets and of ${\mathcal N}=2$ chiral operators. For both
correlation functions we review the solution of the superconformal Ward
identities and describe their superconformal block decompositions. This
provides the foundation for an extensive numerical analysis discussed in the
second half of the paper. We find a large number of constraints for operator
dimensions, OPE coefficients, and central charges that must hold for any
${\mathcal N}=2$ superconformal field theory.
