  A weak wave turbulence theory is developed for two-dimensional (2D)
magnetohydrodynamics (MHD). We derive and analyze the kinetic equation
describing the three-wave interactions of pseudo-Alfv\'en waves. Our analysis
is greatly helped by the fortunate fact that in 2D the wave-kinetic equation is
integrable. In contrast with the 3D case, in 2D the wave interactions are
nonlocal. Another distinct feature is that strong derivatives of spectra tend
to appear in the region of small parallel (i.e. along the uniform magnetic
field direction) wavenumbers leading to a breakdown of the weak turbulence
description in this region. We develop a qualitative theory beyond weak
turbulence describing subsequent evolution and formation of a steady state.
