  We implement the conformal bootstrap for N=4 superconformal field theories in
four dimensions. Consistency of the four-point function of the stress-energy
tensor multiplet imposes significant upper bounds for the scaling dimensions of
unprotected local operators as functions of the central charge of the theory.
At the threshold of exclusion, a particular operator spectrum appears to be
singled out by the bootstrap constraints. For large values of the central
charge, this extremal spectrum is compatible with that of supergravity in AdS_5
x S^5. For finite central charge, we conjecture that the extremal spectrum is
that of N=4 SYM at an S-duality invariant value of the complexified gauge
coupling.
