  Multitudinous combinatorial structures are counted by generating functions
satisfying a composition scheme $F(z)=G(H(z))$. The corresponding asymptotic
analysis becomes challenging when this scheme is critical (i.e., $G$ and $H$
are simultaneously singular). The singular exponents appearing in the Puiseux
expansions of $G$ and $H$ then dictate the asymptotics.
  In this work, we first complement results of Flajolet et al. for a full
family of singular exponents of $G$ and $H$. Motivated by many examples (random
mappings, planar maps, directed lattice paths), we consider a natural extension
of this scheme, namely $F(z,u)=G(u H(z))M(z)$. We also consider a variant of
this scheme, which allows us to analyse the number of $H$-components of a given
size in $F$.
  These two models lead to a rich world of limit laws, where we identify the
key r\^ole played by a new universal three-parameter law: the
beta-Mittag-Leffler distribution, which is essentially the product of a beta
and a Mittag-Leffler distribution. We prove (double) phase transitions,
additionally involving Boltzmann and mixed Poisson distributions, with a
unified explanation of the associated thresholds. We also obtain moment
convergence and local limit theorems. We end with extensions of the critical
composition scheme to a cycle scheme and to the multivariate case, leading to
product distributions. Applications are presented for random walks, trees
(supertrees of trees, increasingly labelled trees, preferential attachment
trees), triangular P\'olya urns, and the Chinese restaurant process.
