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When light and matter are weakly coupled, they can be described as two distinctive systems exchanging quanta of energy. By contrast, for very large coupling strength, the systems hybridize and form compounds that cannot be described in terms of light or matter only. In this Thesis, we study some exotic properties which arise in this regime. In particular, we are interested in the possibility to engineer quantum phase transitions in these systems. One direction we explore is the study of two-photon coupling, a mechanism in which matter creates or absorb photons by pairs. This mechanism creates a rich phase diagram containing both phase transitions and instabilities. A second topic is the use of these transitions for sensing applications. Indeed, near the critical point, the system becomes extremely sensitive to external perturbations. We study a setup in which a single qubit is coupled to a bosonic field. We show that even this finite-size system displays a phase transition, which can be used to measure the frequency of the qubit and the field with improved accuracy. This protocol could be used to develop small-scale sensors. Finally, we study how the ability of a system to perform certain metrological tasks could be used to characterize and quantify nonclassicality, by using the formalism of resource theories. The first three chapters present the main concepts and key results in the fields of ultrastrong light-matter coupling, superradiant phase transitions, and quantum metrology. I have strived to write these chapters pedagogically; they can be used by non-specialists or students as an introduction to these domains. The three other chapters present my own research contributions. Although most of those have already been published elsewhere, this manuscript contains additional results, remarks, and perspectives, and should be considered as an improved version of the original papers.