I discuss two related nonlinear mechanisms of tidal dissipation that require finite tidal deformations for their operation: the elliptical instability and the precessional instability. Both are likely to be important for the tidal evolution of short-period extrasolar planets. The elliptical instability is a fluid instability of elliptical streamlines, such as in tidally deformed non-synchronously rotating or non-circularly orbiting planets. I summarise the results of local and global simulations that indicate this mechanism to be important for tidal spin synchronisation, planetary spin-orbit alignment and orbital circularisation for the shortest period hot Jupiters. The precessional instability is a fluid instability that occurs in planets undergoing axial precession, such as those with spin-orbit misalignments (non-zero obliquities). I summarise the outcome of local MHD simulations designed to study the turbulent damping of axial precession, which suggest this mechanism to be important in driving tidal evolution of the spin-orbit angle for hot Jupiters. Avenues for future work are also discussed
