The Abelian Sandpile Model (ASM) is an archetypical model of the physical phenomenon called self-organized criticality. One of the main open problems about this model, originally defined on the Euclidean lattice, is to provide rigorous mathematical explanation for predictions about the values of its critical exponents, originating in physics. The present thesis is devoted to the study of the ASM on a large class of randomly rooted graphs of self-similar nature which are realized as Schreier graphs of actions on rooted trees, by automorphisms, of iterated monodromy groups associated with complex polynomials. The main result is the computation of the critical exponent related to the mass of avalanches of sand of the model on these graphs, proving thus the critical behaviour of the ASM.