  The calculation of the two-loop corrections to the three-jet production rate
and to event shapes in electron--positron annihilation requires the computation
of a number of two-loop four-point master integrals with one off-shell and
three on-shell legs. Up to now, only those master integrals corresponding to
planar topologies were known. In this paper, we compute the yet outstanding
non-planar master integrals by solving differential equations in the external
invariants which are fulfilled by these master integrals. We obtain the master
integrals as expansions in $\e=(4-d)/2$, where $d$ is the space-time dimension.
The fully analytic results are expressed in terms of the two-dimensional
harmonic polylogarithms already introduced in the evaluation of the planar
topologies.
