In recent years there has been growing interest in open quantum systems described by non-Hermitian Hamiltonians in various fields. In this talk I present results on the quantum evolution of Gaussian wave packets generated by a non-Hermitian Hamiltonian in the semiclassical limit of small $\hbar$. This yields a generalisation of the Ehrenfest theorem for the dynamics of observable expectation values. The resulting equations of motion for dynamical variables are coupled to an equation of motion for the phase-space metric — a phenomenon having no analogue in Hermitian theories. The insight that can be gained by this classical description will be demonstrated for a number of example systems.