  Topological superconductors admit zero-energy Majorana bound states at their
boundaries. In this review article, we discuss how to probe these Majorana
bound states in Josephson junctions between two topological superconductors. In
the absence of an applied bias, the presence of these states gives rise to an
Andreev bound state whose energy varies $4\pi$-periodically in the
superconducting phase difference. An applied voltage bias leads to a
dynamically varying phase according to the Josephson relation. Furthermore, it
leads to dynamics of the occupation of the bound state via its non-adiabatic
coupling to the continuum. While the Josephson relation suggests a fractional
Josephson effect due to the $4\pi$-periodicity of the bound state, its
observability relies on the conservation of the occupation of the bound state
on the experimentally probed time scale. We study the lifetime of the bound
state and identify the time scales it has to be compared to. In particular, we
are interested in signatures of the fractional Josephson effect in the Shapiro
steps and in current noise measurements. We also discuss manifestations of the
zero-energy Majorana states on the dissipative subgap current.
