Let $T$ be a standard torus in $S^3$. $K$ is a $(1,1)$-knot if $K$ can be positioned so that $K$ intersects $T$ in two points, which divide $K$ into two arcs, and such that each of the arcs is parallel to a simple arc lying on $T$.

We give a description of all $(1,1)$-knots which contain an essential meridional surface, that is, an incompressible, meridionally incompressible, not $\partial$-parallel, properly embedded surface in the exterior of a knot $K$, whose boundary consists of meridians of $K$.

In particular, we show that for given $g > 0$ and $h > 0$, there are $(1,1)$-knots which contain an essential meridional surface of genus $g$, and whose boundary has $2h$ components. This contrasts with a result of Gordon and Reid, which shows that $(1,1)$-knots cannot contain "planar" essential meridional surfaces.