This book is devoted to the theory and applications of second-order necessary and sufficient optimality conditions in the calculus of variations and optimal control. The authors develop theory for a control problem with ordinary differential equations subject to boundary conditions of equality and inequality type, and for mixed state-control constraints of equality type. The book has several distinctive features: necessary and sufficient conditions are given in the form of no-gap conditions; the theory covers broken extremals where the control has finitely many points of discontinuity; and a number of numerical examples in various areas of application are fully solved. This book will be of interest to academic researchers in calculus of variations and optimal control. It will also be a useful resource to researchers and engineers who use applications of optimal control in areas such as mechanics, mechatronics, physics, economics, or chemical, electrical and biological engineering.