  We study free boundary minimal surfaces in the unit ball of low
cohomogeneity. For each pair of positive integers $(m,n)$ such that $m, n >1$
and $m+n\geq 8$, we construct a free boundary minimal surface $\Sigma_{m, n}
\subset B^{m+n}$(1) invariant under $O(m)\times O(n)$. When $m+n<8$, an
instability of the resulting equation allows us to find an infinite family
$\{\Sigma_{m,n, k}\}_{k\in \mathbb{N}}$ of such surfaces. In particular,
$\{\Sigma_{2, 2, k}\}_{k\in \mathbb{N}}$ is a family of solid tori which
converges to the cone over the Clifford Torus as $k$ goes to infinity. These
examples indicate that a smooth compactness theorem for Free Boundary Minimal
Surfaces due to Fraser and Li does not generally extend to higher dimensions.
  For each $n\geq 3$, we prove there is a unique nonplanar $SO(n)$-invariant
free boundary minimal surface (a "catenoid") $\Sigma_n \subset B^n(1)$. These
surfaces generalize the "critical catenoid" in $B^3(1)$ studied by Fraser and
Schoen.
