For many types of random planar maps, i.e. planar graphs embedded in the sphere, it is known that their geometry possesses a scaling limit described by a universal random continuous metric space known as the Brownian sphere. I will give a short overview of its properties and its relation to Liouville Quantum Gravity and discuss ways to escape the universality class by introducing matter degrees of freedom or vertices with heavy-tailed degree distribution. The peeling process will be introduced as a general method to explore planar maps, and can be used to study the geometry of models in the Brownian sphere universality class and beyond.