  Open many-body quantum systems have attracted renewed interest in the context
of quantum information science and quantum transport with biological clusters
and ultracold atomic gases. The physical relevance in many-particle bosonic
systems lies in the realization of counter-intuitive transport phenomena and
the stochastic preparation of highly stable and entangled many-body states due
to engineered dissipation. We review a variety of approaches to describe an
open system of interacting ultracold bosons which can be modeled by a
tight-binding Hubbard approximation. Going along with the presentation of
theoretical and numerical techniques, we present a series of results in diverse
setups, based on a master equation description of the dissipative dynamics of
ultracold bosons in a one-dimensional lattice. Next to by now standard
numerical methods such as the exact unravelling of the master equation by
quantum jumps for small systems and beyond mean-field expansions for larger
ones, we present a coherent-state path integral formalism based on
Feynman-Vernon theory applied to a many-body context.
