  Recent advances in nuclear structure theory have significantly enlarged the
accessible part of the nuclear landscape via ab initio many-body calculations.
These developments open new ways for microscopic studies of light, medium-mass
and heavy nuclei as well as nuclear matter and represent an important step
toward a systematic and comprehensive understanding of atomic nuclei across the
nuclear chart. While remarkable agreement has been found between different
many-body methods for a given nuclear Hamiltonian, the comparison with
experiment and the understanding of theoretical uncertainties are still
important open questions. The observed discrepancies to experiment indicate
deficiencies in presently used nuclear interactions and operators. Chiral
effective field theory (EFT) allows to systematically derive contributions to
nucleon-nucleon (NN), three-nucleon (3N) and higher-body interactions including
estimates of theoretical uncertainties. While the treatment of NN interactions
in many-body calculations is well established, the calculation of 3N
interactions and their incorporation in ab initio frameworks is still a
frontier. This work reviews in detail recent and current developments on the
derivation and implementation of improved 3N interactions and provides a
comprehensive introduction to fundamental methods for their practical
calculation and representation. We further give an overview of novel and
established methods that facilitate the inclusion and treatment of 3N
interactions in ab initio nuclear structure frameworks and present a selection
of the latest calculations of atomic nuclei as well as nuclear matter based on
state-of-the-art nuclear NN and 3N interactions derived within chiral EFT.
Finally, we discuss ongoing efforts, open questions and future directions.
