The project "MUSTANG" (funded by the French national research agency) aims at studying two, related, random objects: multitype spatial trees and maps, with a particular emphasis on large degree regimes that fall outside the Brownian universality classes and relate to statistical physics models. We consider both the questions of scaling limits of discrete models to continuum ones and the properties of theses limits (their singular geometry, fractal properties). This brings together techniques from combinatorics (bijections and analytic combinatorics) and probability (random graphs, discrete and continuous stochastic processes, especially Gaussian and Lévy processes).