  The cosmic large-scale structures of the Universe are mainly the result of
the gravitational instability of initially small density fluctuations in the
dark-matter distribution. Dark matter appears to be initially cold and behaves
as a continuous and collisionless medium on cosmological scales, with evolution
governed by the gravitational Vlasov--Poisson equations. Cold dark matter can
accumulate very efficiently at focused locations, leading to a highly
non-linear filamentary network with extreme matter densities. Traditionally,
investigating the non-linear Vlasov--Poisson equations was typically reserved
for massively parallelised numerical simulations. Recently, theoretical
progress has allowed us to analyse the mathematical structure of the first
infinite densities in the dark-matter distribution by elementary means. We
review related advances, as well as provide intriguing connections to classical
plasma problems, such as the beam-plasma instability.
