  We compute exact 2- and 3-point functions of chiral primaries in
four-dimensional N=2 superconformal field theories, including all perturbative
and instanton contributions. We demonstrate that these correlation functions
are nontrivial and satisfy exact differential equations with respect to the
coupling constants. These equations are the analogue of the $tt^*$ equations in
two dimensions. In the SU(2) N=2 SYM theory coupled to 4 hypermultiplets they
take the form of a semi-infinite Toda chain. We provide the complete solution
of this chain using input from supersymmetric localization. To test our results
we calculate the same correlation functions independently using Feynman
diagrams up to 2-loops and we find perfect agreement up to the relevant order.
As a spin-off, we perform a 2-loop check of the recent proposal of
arXiv:1405.7271 that the logarithm of the sphere partition function in N=2
SCFTs determines the K\"ahler potential of the Zamolodchikov metric on the
conformal manifold. We also present the $tt^*$ equations in general SU(N) N=2
superconformal QCD theories and comment on their structure and implications.
