The quintom theory of dark energy after DESI DR2

Observations from the Dark Energy Spectroscopic Instrument Data Release 2 (DESI DR2) challenge the [Formula: see text] cold dark matter ([Formula: see text]CDM) paradigm by suggesting that the equation-of-state parameter of dark energy evolves across [Formula: see text], a phenomenon known as the quintom scenario. Motivated by this development, we present a staged review of quintom cosmology, including its theoretical foundations, observational supports, implications and possible extensions. We first trace the historical progression from Einstein's static cosmological constant to modern dynamical dark energy, summarizing recent cosmological constraints that favor an evolving [Formula: see text] with time. A key focus is the theoretical no-go theorem for dark energy, which shows that no single canonical field or perfect-fluid model can smoothly cross the [Formula: see text] boundary. We then survey viable quintom constructions, including two-field models, single-scalar-field models with higher derivatives, modified-gravity frameworks, interacting dark energy and effective field theory approaches that unify these mechanisms. Possible interactions between quintom fields and ordinary matter, as well as their potential roles in yielding non-singular universe solutions, are discussed.