  We construct a family of compact free boundary minimal annuli immersed in the
unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$, the first such examples other than
the critical catenoid. This solves a problem formulated by Nitsche in 1985.
These annuli are symmetric with respect to two orthogonal planes and a finite
group of rotations around an axis, and are foliated by spherical curvature
lines. We show that the only free boundary minimal annulus embedded in
$\mathbb{B}^3$ foliated by spherical curvature lines is the critical catenoid;
in particular, the minimal annuli that we construct are not embedded. On the
other hand, we also construct families of non-rotational compact embedded
capillary minimal annuli in $\mathbb{B}^3$. Their existence solves in the
negative a problem proposed by Wente in 1995.

Differential Geometry