### Abstract

The dynamics of the asymmetric exclusion process is governed by the spectrum of its transition matrix. In particular its lowest excited state describes the approach to stationarity at large times. I will discuss the exact diagonalisation of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. The resulting Bethe ansatz equations describe the {\em complete} spectrum of the transition matrix. For totally asymmetric diffusion I will present exact results for the spectral gap and derive the dynamical phase diagram. We observe boundary induced crossovers in and between massive, diffusive and KPZ scaling regimes.