A long-standing goal of nuclear theory is to determine the properties of atomic nuclei based on the fundamental interactions among the protons and neutrons (i.e., nucleons). By adopting nucleon-nucleon (NN), three-nucleon (NNN) and higher-nucleon interactions determined from either meson-exchange theory or QCD, with couplings fixed by few-body systems, we preserve the predictive power of nuclear theory. This foundation enables tests of nature's fundamental symmetries and offers new vistas for the full range of complex nuclear phenomena. Basic questions that drive our quest for a microscopic predictive theory of nuclear phenomena include: (1) What controls nuclear saturation; (2) How the nuclear shell model emerges from the underlying theory; (3) What are the properties of nuclei with extreme neutron/proton ratios; (4) Can we predict useful cross sections that cannot be measured; (5) Can nuclei provide precision tests of the fundamental laws of nature; and (6) Under what conditions do we need QCD to describe nuclear structure, among others. Along with other ab initio nuclear theory groups, we have pursued these questions with meson-theoretical NN interactions, such as CD-Bonn and Argonne V18, that were tuned to provide high-quality descriptions of the NN scattering phase shifts and deuteron properties. We then add meson-theoretic NNN interactions such as the Tucson-Melbourne or Urbana IX interactions. More recently, we have adopted realistic NN and NNN interactions with ties to QCD. Chiral perturbation theory within effective field theory ([chi]EFT) provides us with a promising bridge between QCD and hadronic systems. In this approach one works consistently with systems of increasing nucleon number and makes use of the explicit and spontaneous breaking of chiral symmetry to expand the strong interaction in terms of a dimensionless constant, the ratio of a generic small momentum divided by the chiral symmetry breaking scale taken to be about 1 GeV/c. The resulting NN and NNN interactions, characterized by the order of the expansion retained (e.g. 'next-to-next-to leading order' is NNLO), provide a high-quality fit to the NN data and the A = 3 ground-state (g.s.) properties. The derivations of NN, NNN, etc. interactions within meson-exchange and [chi]EFT are well-established but are not subjects of this review. Our focus is solution of the non-relativistic quantum many-body Hamiltonian that includes these interactions using our no core shell model (NCSM) formalism. In the next section we will briefly outline the NCSM formalism and then present applications, results and extensions in later sections.