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, defects, source terms, and cracks. Over the last decade, topological asymptotic analysis has become a broad, rich, and fascinating research area from both theoretical and numerical standpoints. It has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing, and mechanical modeling, including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis, and damage evolution modeling. Since there is no monograph on the subject at present, the authors provide here the first account of the theory which combines classical sensitivity analysis in shape optimization.