The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, source- terms and cracks. This relatively new concept has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing, multi- scale material design and mechanical modeling including damage and fracture evolution phenomena. In this talk, the topological derivative method is presented, together with a portfolio of applications in the context of topology optimization, inverse problems and fracture mechanics.