 Even if genuinely hyperintensional language is accepted into our philosophical theorizing, this leaves many further issues open. One pressing one is the question of “just ‘how hyper’ hyperintensions are” (Jespersen &amp; Duží 2015: 527): what new distinctions should our theorizing recognize beyond necessary equivalence? The mathematical structure of the realm of hyperintensionality is not well understood yet. We come to this in  Section 3.  However, the early twenty-first century is seeing a hyperintensional revolution (Nolan 2014), with different general approaches currently being developed, which qualify as hyperintensional in some sense. We provide a critical survey in  Section 4,  where we also assess the approaches in terms of how they fare with respect to hyperintensional phenomena singled out in previous Sections.