  The two-dimensional case occupies a special position in the theory of
critical phenomena due to the exact results provided by lattice solutions and,
directly in the continuum, by the infinite-dimensional character of the
conformal algebra. However, some sectors of the theory, and most notably
criticality in systems with quenched disorder and short range interactions,
have appeared out of reach of exact methods and lacked the insight coming from
analytical solutions. In this article we review recent progress achieved
implementing conformal invariance within the particle description of field
theory. The formalism yields exact unitarity equations whose solutions classify
critical points with a given symmetry. It provides new insight in the case of
pure systems, as well as the first exact access to criticality in presence of
short range quenched disorder. Analytical mechanisms emerge that in the random
case allow the superuniversality of some critical exponents and make explicit
the softening of first order transitions by disorder.
