  We develop the conformal bootstrap program for six-dimensional conformal
field theories with $(2,0)$ supersymmetry, focusing on the universal four-point
function of stress tensor multiplets. We review the solution of the
superconformal Ward identities and describe the superconformal block
decomposition of this correlator. We apply numerical bootstrap techniques to
derive bounds on OPE coefficients and scaling dimensions from the constraints
of crossing symmetry and unitarity. We also derive analytic results for the
large spin spectrum using the lightcone expansion of the crossing equation. Our
principal result is strong evidence that the $A_1$ theory realizes the minimal
allowed central charge $(c=25)$ for any interacting $(2,0)$ theory. This
implies that the full stress tensor four-point function of the $A_1$ theory is
the unique unitary solution to the crossing symmetry equation at $c=25$. For
this theory, we estimate the scaling dimensions of the lightest unprotected
operators appearing in the stress tensor operator product expansion. We also
find rigorous upper bounds for dimensions and OPE coefficients for a general
interacting $(2,0)$ theory of central charge $c$. For large $c$, our bounds
appear to be saturated by the holographic predictions obtained from
eleven-dimensional supergravity.
