We pose and solve an ’inverse optimal’ adaptive tracking problem for non-linear systems with unknown parameters. A controller is said to be inverse optimal when it minimizes a meaningful cost functional that incorporates integral penalty on the tracking error state and the control, as well as a terminal penalty on the parameter estimation error. The basis of our method is an adaptive tracking control Lyapunov function (atclf) the existence of which guarantees the solvability of the inverse optimal problem. The controllers designed in this paper are not of certainty-equivalence type. Even in the linear case they would not be a result of solving a Riccati equation for a given value of the parameter estimate. Our abandoning of the certainty-equivalence approach is motivated by the fact that, in general, this approach does not lead to optimality of the controller with respect to the overall plant-estimator system, even though both the estimator and the controller may be optimal as separate entities. Our controllers, instead, compensate for the effect of parameter adaptation transients in order to achieve optimality of the overall system. We combine inverse optimality with backstepping to design a new class of adaptive controllers for strict-feedback systems. These controllers solve a problem left open in the previous adaptive backstepping designs, i.e. obtaining transient performance bounds that include an estimate of control effort, which is the first such result in the adaptive control literature.