  Motivated by applications to critical phenomena and open theoretical
questions, we study conformal field theories with $O(m)\times O(n)$ global
symmetry in $d=3$ spacetime dimensions. We use both analytic and numerical
bootstrap techniques. Using the analytic bootstrap, we calculate anomalous
dimensions and OPE coefficients as power series in $\varepsilon=4-d$ and in
$1/n$, with a method that generalizes to arbitrary global symmetry. Whenever
comparison is possible, our results agree with earlier results obtained with
diagrammatic methods in the literature. Using the numerical bootstrap, we
obtain a wide variety of operator dimension bounds, and we find several islands
(isolated allowed regions) in parameter space for $O(2)\times O(n)$ theories
for various values of $n$. Some of these islands can be attributed to fixed
points predicted by perturbative methods like the $\varepsilon$ and large-$n$
expansions, while others appear to arise due to fixed points that have been
claimed to exist in resummations of perturbative beta functions.
