  Bohmian mechanics provides an explanation of quantum phenomena in terms of
point particles guided by wave functions. This review focuses on the formalism
of non-relativistic Bohmian mechanics, rather than its interpretation. Although
the Bohmian and standard quantum theories have different formalisms, both give
exactly the same predictions for all phenomena. Fifteen years ago, the quantum
chemistry community began to study the practical usefulness of Bohmian
mechanics. Since then, the scientific community has mainly applied it to study
the (unitary) evolution of single-particle wave functions, either by developing
efficient quantum trajectory algorithms or by providing a trajectory-based
explanation of complicated quantum phenomena. Here we present a large list of
examples showing how the Bohmian formalism provides a useful solution in
different forefront research fields for this kind of problems (where the
Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this
work also emphasizes that the Bohmian formalism can be a useful tool in other
types of (non-unitary and nonlinear) quantum problems where the influence of
the environment or the global wave function are unknown. This review contains
also examples on the use of the Bohmian formalism for the many-body problem,
decoherence and measurement processes. The ability of the Bohmian formalism to
analyze this last type of problems for (open) quantum systems remains mainly
unexplored by the scientific community. The authors of this review are
convinced that the final status of the Bohmian theory among the scientific
community will be greatly influenced by its potential success in these type of
problems that present non-unitary and/or nonlinear quantum evolutions. A brief
introduction of the Bohmian formalism and some of its extensions are presented
in the last part of this review.
