  We present a comprehensive review of the In-Medium Similarity Renormalization
Group (IM-SRG), a novel ab inito method for nuclei. The IM-SRG employs a
continuous unitary transformation of the many-body Hamiltonian to decouple the
ground state from all excitations, thereby solving the many-body problem.
Starting from a pedagogical introduction of the underlying concepts, the IM-SRG
flow equations are developed for systems with and without explicit spherical
symmetry. We study different IM-SRG generators that achieve the desired
decoupling, and how they affect the details of the IM-SRG flow. Based on
calculations of closed-shell nuclei, we assess possible truncations for closing
the system of flow equations in practical applications, as well as choices of
the reference state. We discuss the issue of center-of-mass factorization and
demonstrate that the IM-SRG ground-state wave function exhibits an approximate
decoupling of intrinsic and center-of-mass degrees of freedom, similar to
Coupled Cluster (CC) wave functions. To put the IM-SRG in context with other
many-body methods, in particular many-body perturbation theory and
non-perturbative approaches like CC, a detailed perturbative analysis of the
IM-SRG flow equations is carried out. We conclude with a discussion of ongoing
developments, including IM-SRG calculations with three-nucleon forces, the
multi-reference IM-SRG for open-shell nuclei, first non-perturbative
derivations of shell- model interactions, and the consistent evolution of
operators in the IM-SRG. We dedicate this review to the memory of Gerry Brown,
one of the pioneers of many-body calculations of nuclei.
