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We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models.For d < 1 matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising-like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces. 6/93, submitted to Physics Reports Contents

We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.In a map, the valence or degree of a face or vertex is the number of its incident edges 4 .Maps with only 4-valent faces are called quadrangulations which are dual to tetravalent maps where all vertices have degree 4. This class of maps is the one for which bijections are the simplest. There are actually two of them [5]: one between tetravalent maps and so-called blossom trees, the other between quadrangulations and well-labeled trees.So far, generalizations to wider classes of maps were obtained mainly in terms of blossom trees. A first extension consists of enumerating maps with prescribed vertex valences. The corresponding generating functions are easily derived by considering the general "one-matrix model", or by recursive decomposition [12], while the corresponding bijective proof involves blossom trees with subtle charge constraints [13]. This is drastically simplified in the case of maps with only vertices of even valence, corresponding to even potentials in the matrix model formulation. The corresponding blossom trees [14] then have a simple characterization which makes it possible to re-derive Tutte's compact formulas for the numbers of maps with prescribed (even) vertex valences [15].More generally, one may as well try to enumerate the bipartite, i.e. vertex-bicolored, maps with prescribed vertex valences of either color, which corresponds to the general "two-matrix model". This problem has many physical applications, including, for instance, the celebrated Ising model on random lattices [16] as well as a whole range of multicritical theories corresponding to minimal models of CFT coupled to 2D quantum gravity [17]. The bijective enumeration via blossom trees was found by .This indeed extends the previous case, as arbitrary maps are equivalent to bipartite maps with only two-valent black vertices. Another interesting subcase is that of p-constellations with only p-valent black vertices and white vertices with valences that are multiples of p [19]. This corresponds to the most general situation where explicit compact formulas [20] are known for the numbers of maps with prescribed vertex valences. To complete the picture, note that 2-constellations are equivalent to maps with even vertex valences.On the other (dual) front, well-labeled trees appeared so far only in correspondence with quadrangulations [5] [9] and Eulerian (fac...