We introduce fundamental gauge theories that can be employed to construct informed composite bright and dark extensions of the Standard Model, within and beyond the standard paradigms. The gap between theory and experiments is bridged by providing predictions and ways to test them, for example, at the Fermi scale and via precision flavor experiments. We will review time-honored paradigms from (walking) technicolor to composite Goldstone Higgs and discuss their features and differences. Standard Model fermion mass generation in composite models will also be discussed along with the challenges and opportunities that it offers. To be concrete and pedagogical we will concentrate on minimal constructions featuring strongly coupled gauge theories supporting the global symmetry breaking pattern SU(4)/Sp(4). The most minimal underlying fundamental description consists of an SU(2) gauge theory with two Dirac fermions transforming according to the fundamental representation of the gauge group. This minimal choice enables us to use first principle lattice results to predict the massive spectrum for models of composite (Goldstone) Higgs dynamics and strongly interacting dark matter, of immediate impact for current and future experimental searches. Because composite dynamics embraces a rich spectrum of theories with dynamics ranging from QCD-like behavior to (near) conformal one, we also report here the state-of-the-art of numerical and analytic properties of several strongly coupled theories including their spectrum, phase diagrams and, when applicable, their (near) conformal data.