In this talk, we will discussion the interest and efficiency of subdivision solvers in geometric problems. We will first consider the problem of isolation of intersection points, or equivalently the resolution of polynomial equations. We will describe a new method, which improves substantially previous method such as Interval Projection Polyhedron method and illustrate its behavior on problems from Computer Aided Geometric Design. Next, we will focus on computing the topology of implicit curves and surfaces. We will describe a method, which allows us to mesh with guarantees, smooth implicit surface. A special attention will be given to the treatment of singularities, based on stratified Morse theory.