  We combine supersymmetric localization and the conformal bootstrap to study
five-dimensional superconformal field theories. To begin, we classify the
admissible counter-terms and derive a general relation between the five-sphere
partition function and the conformal and flavor central charges. Along the way,
we discover a new superconformal anomaly in five dimensions. We then propose a
precise triple factorization formula for the five-sphere partition function,
that incorporates instantons and is consistent with flavor symmetry
enhancement. We numerically evaluate the central charges for the rank-one
Seiberg and Morrison-Seiberg theories, and find strong evidence for their
saturation of bootstrap bounds, thereby determining the spectra of long
multiplets in these theories. Lastly, our results provide new evidence for the
$F$-theorem and possibly a $C$-theorem in five-dimensional superconformal
theories.
