The Standard Model of particle physics stands as the most accurate description we have of our observed phenomena. It accommodates the experimental data collected to date, and provides an economical and predictive framework for understanding nature on small scales. The model can in fact be consistently extrapolated to the smallest length scales we can imagine, where the concept of spacetime itself is believed to require modification. As such, the Standard Model stands as a truly monumental achievement of scientific pursuits.

The successes of the model, however, present some equally profound questions. The model, for instance, can in fact be extrapolated to very high scales, but only at the cost of introducing a highly unstable hierarchy. This thesis addresses the possibilities afforded by conformal field theories in addressing this problem. There are three classes of models discussed: four-dimensional composite models, five-dimensional composite models, and unparticle models. The foundations of each scenario are reviewed, and new approaches to solving some of their problems are presented. In each case, conformality plays a central role. Generically this is due to the fact that nontrivial scale dependence is at the heart of conformal field theories: we will see the common occurrence of large anomalous dimensions in strongly-coupled conformal field theories, which allow a softening of the dependence of relatively low-energy physics on unknown physics at higher energies.

Finding a mechanism to achieve a stable separation of low-scale physics presents many challenges, but typically gives concrete predictions for new phenomena to be observed at the Large Hadron Collider, which recently became the world's highest energy particle accelerator. What will be revealed there in coming years is still very uncertain, so knowing in advance how each theoretical construction will be manifested physically is the immediate concern of particle physicists. The goal of this work is to show how the use of two very powerful tools—strong coupling and conformality in four and five dimensions—can vary within different models, how these variations help solve problems facing the Standard Model, and how different models can lead to clear signatures in forthcoming observations.