The φ 4 model has been the "workhorse" of the classical Ginzburg-Landau phenomenological theory of phase transitions and, furthermore, the foundation for a large amount of the now-classical developments in nonlinear science. However, the φ 4 model, in its usual variant (symmetric double-well potential), can only possess two equilibria. Many complex physical systems possess more than two equilibria and, furthermore, the number of equilibria can change as a system parameter (e.g., the temperature in condensed matter physics) is varied. Thus, "higherorder field theories" come into play. This chapter discusses recent developments of higher-order field theories, specifically the φ 6 , φ 8 models and beyond. We first establish their context in the Ginzburg-Landau theory of successive phase transitions, including a detailed discussion of the symmetric triple well φ 6 potential and its properties. We also note connections between field theories in high-energy physics (e.g., "bag models" of quarks within hadrons) and parametric (deformed) φ 6 models. We briefly mention a few salient points about even-higher-order field theories of the φ 8 , φ 10 , etc. varieties, including the existence of kinks with power-law tail asymptotics that give rise to long-range interactions. Finally, we conclude with a set of open problems in the context of higher-order scalar fields theories. arXiv:1806.06693v3 [hep-th]