Legged robots do not require a flat and smooth surface to advance and have, therefore, thepotential to work in unstructured and complex environments. These robots, like humansand animals, can climb stairs and ladders and cross challenging terrain, which is essentialfor exploration and rescue applications.However, walking is a complex task. The robot movement must be a consequence of creat-ing contacts with the environment. Every time a new contact is made the dynamics becomediscontinuous. Also, legged robots are unstable and subject to many constraints, which re-stricts the potential movements. Finally, contact planning has a combinatorial structure: therobot chooses the optimal subset of contacts out of an infinite number of possible contactpoints.Trajectory optimization in robotics is often decoupled into two independent modules. Aproblem formulation, that models the robot motion and transforms it into a mathematicaloptimization problem, and the resolution of this optimization problem with a suitable op-timization solver. We rather envision this question as a two-way interaction between thetwo modules, that can not be considered separately. Indeed, the robotics problem formu-lation must also be a consequence of the understanding of the capabilities of the numericaloptimization algorithm.We propose a simultaneous trajectory and contact optimization with an augmented La-grangian algorithm. The contacts with the environment are modeled in a continuous waywith complementarity constraints. For example, the normal contact model is defined bytwo inequalities: positive force and distance to the surface, and a complementarity relation:either the force or the distance to the surface is zero.These constraints are added as path constraints in a continuous optimal control formula-tion. Using a direct collocation, the formulation is converted into a non-linear constrainedoptimization problem. Optimization problems with complementarity constraints have a de-generated and challenging structure and do not fulfil basic constraints qualifications usedby standard numerical solvers. Therefore, we propose to solve it using an augmented La-grangian algorithm, which offers a good behaviour under this type of constraints and canbe efficiently warmstarted.In this thesis we present our preliminary results with a robot manipulator interacting withthe environment. Optimizing simultaneously trajectory and contacts, we compute interest-ing motions such as multiple surface touching and pick and place tasks. We also outline ourfuture plans to generate walking motions with legged robots.