While the Standard Model of particle physics has undoubtedly been an experimental success, several questions remain unresolved. In particular, the Standard Model cannot account for the observed cosmological preference for matter over dark matter, nor does it provide a viable candidate for dark matter. This motivates us to consider extensions to the Standard Model; in this thesis, we will focus on several extensions of the Standard Model in which the formation of bound states is a significant factor. We will argue that the formation of bound states produces new phenomena that can address these unsettled questions.

First, we consider a strongly-coupled version of the Minimal Supersymmetric Standard Model. We demonstrate that in this model, electroweak symmetry breaking may be triggered by the presence of squark bound states which mix with the fundamental Higgs boson. Next, we show that this model has a viable phenomenology (e.g., it does not have large flavor-changing-neutral-currents or break SUC(3) symmetry). Additionally, this strongly-coupled version of the MSSM can relatively easily accommodate electroweak scale baryogenesis.

Following this, we turn our attention to the possibility of dark matter bound states in asymmetric dark matter models. We first consider a simplistic scalar model and demonstrate that bound state formation can produce a detectable gamma ray excess in certain regions of parameter space. This signal is produced through the decay of the dark force mediator whose emission necessarily accompanies bound state formation. Next, we consider models in which the dark matter self-interactions are described by a broken UD (1) gauge group. We argue that in such models dark matter is generically multi-component, consisting of two species of ions along with dark atoms. We then investigate the possibility of using these self-interactions between the different species to alleviate tension between the cold dark matter paradigm and observations of dwarf galaxies, while retaining the ellipticity of larger halos.

Finally, we consider the formation and growth of Q-balls (non-topological solitons) in a simplified model inspired by the MSSM. In particular models, Q-balls can trigger a phase transition once they reach a critical size. In certain regions of parameter space, small charge Q-balls can be approximated using the Bethe-Salpeter equation. This allows us to study the growth of small Q-balls; by joining this to the semi-classical regime at large charges, we can analyze their growth from individual squarks to critical size. In our simplistic model, we show that Q-balls can indeed reach critical size on cosmological time scales.