In this talk I will present several classes of fractals of overlapping construction, which appear naturally in non-invertible smooth dynamical systems, and in iterated function systems. The study of these fractals is very different from those obtained from diffeomorphisms or iterated function systems with open set condition. This is due to the variable number of preimages, or to the image discs which intersect over the points. I will give several recent results about the behavior of such fractals, from the point of view of dimension, and also from an ergodic point of view.