Complementarity constraints are a very versatile modeling tool, widely used in important economics and engineering areas such as robotics, virtual reality, structural engineering, chemical engineering, data mining, homeland security, nuclear engineering, and network design. Their most common occurrence is in game theoretical models and in hierarchical optimization problems, where the lower-level problem has inequality constraints. Applications involving complementarity constraints span a broad spectrum of models that include contact between rigid or elastic bodies; electricity, commodity or financial markets, and switches in electrical networks. The talk will present recent advances in the investigation of mathematical problems with complementarity constraints and differential problems with complementarity constraints, and will outline outstanding analytical, methodological and computational issues that are of current interest to the optimization and applications communities.