I discuss work done in the last few years, together with my collaborators, in classifying, constructing and studying models of partial compositeness based on underlying 4D gauge theories with fermionic matter. The main motivation is trying to solve the hierarchy problem of the standard model. I start with a brief overview of the main ideas, present one particular example based on a $SU(4)$ gauge group and then discuss more recent work on the dynamics of some neutral pseudo-Nambu-Goldstone bosons that are ubiquitous in all of these models and that could be a target for coming LHC runs. I conclude with some general remarks and future directions.