The physics of the pseudogap phase of high-temperature cuprate superconductors has been an enduring mystery over the past 30 years. The ubiquitous presence of the pseudogap phase in underdoped cuprates suggests that understanding it is key to unraveling the origin of high-temperature superconductivity. We review various theoretical approaches to this problem, emphasizing the concept of emergent symmetries in the underdoped region of those compounds. We differentiate these theories by considering a few fundamental questions related to the rich phenomenology of these materials. Lastly, we discuss a recent idea regarding two kinds of entangled preformed pairs that open a gap at the pseudogap onset temperature, T * , through a specific Higgs mechanism. We review the experimental consequences of this line of thought.