  Alfv\'{e}n waves as excited in black hole accretion disks and neutron star
magnetospheres are the building blocks of turbulence in relativistic,
magnetized plasmas. A large reservoir of magnetic energy is available in these
systems, such that the plasma can be heated significantly even in the weak
turbulence regime. We perform high-resolution three-dimensional simulations of
counter-propagating Alfv\'{e}n waves, showing that an $E_{B_{\perp}}(k_{\perp})
\propto k_{\perp}^{-2}$ energy spectrum develops as a result of the weak
turbulence cascade in relativistic magnetohydrodynamics and its infinitely
magnetized (force-free) limit. The plasma turbulence ubiquitously generates
current sheets, which act as locations where magnetic energy dissipates. We
show that current sheets form as a natural result of nonlinear interactions
between counter-propagating Alfv\'{e}n waves. These current sheets form due to
the compression of elongated eddies, driven by the shear induced by growing
higher order modes, and undergo a thinning process until they break-up into
small-scale turbulent structures. We explore the formation of {current sheets}
both in overlapping waves and in localized wave packet collisions. The
relativistic interaction of localized Alfv\'{e}n waves induces both Alfv\'{e}n
waves and fast waves and efficiently mediates the conversion and dissipation of
electromagnetic energy in astrophysical systems. Plasma energization through
reconnection in current sheets emerging during the interaction of Alfv\'{e}n
waves can potentially explain X-ray emission in black hole accretion coronae
and neutron star magnetospheres.
