We build robots that are meant to look and work like humans, with humans, inspired by humans. But many are the human characteristics that we have not yet understood, as humans are highly complex systems. One fundamental characteristic is compliance, which characterizes human movements. If our body was completely rigid, we would not be able to climb up trees or walk on mountainous paths as easily as we do. But despite being inspired to be a copy of human beings, humanoid robots had rigid links connected with rigid joints since their first appearance. It is only recently that they started to be more “human-like”, with the development of compliant actuators. In this thesis the objective is to analyze of the role of compliance in human walking and in humanoid robots motions. We model both the human body and humanoid robots as rigid multi-body systems. Both systems are highly redundant, reason for which optimization represents an essential tool to achieve our goals. In particular, we adopt optimal control approaches. In many state of the art compliant walking mechanisms, compliance is introduced at joint level by means of elastic components with constant stiffness, due to the difficulty of varying stiffness and the considerable dimensions of currently available variable stiffness actuators. This is the reason for which many studies focused on finding constant joint stiffness during human walking. However, biomechanics studies have shown that stiffness changes in human joints during movements. The questions we want to address are therefore: how does stiffness modulate during human walking and what is the influence of such modulations on the gait? To answer these questions, we used walking motions from motion capture data and a 2D dynamic model of the human body, where the actuation of the leg joints are modeled with torsional springs and bi-articular coupling springs with variable stiffness. We computed the stiffness profiles of these springs, which showed how stiffness changes over the walking cycle and can also assume big values, contrasting with many state of the art walking mechanisms. We proceeded by analyzing how walking gaits are modified if the stiffness modulation is reduced. This further step showed that the original walking gait could be approximated in unconstrained walking scenarios such as level ground and slopes but not in constraint ones as stairs. This result demonstrated the importance of stiffness modulation during walking and can serve for future compliant actuators design. There are several existing humanoid robots with compliant actuators. Among these, the iCub is a widely spreaded advanced research humanoid that has recently acquired legs with Series Elastic Actuators (SEA). The reduced version of it, HeiCub, was delivered to Heidelberg University by the end of 2014 and is the robot used in this thesis. We first analyzed the motion of squatting. The problem is formulated as an optimal control problem where only the three pitch joints of the legs are considered active and the whole-body dynamics of the robot is used. Squat motions for different objective functions are generated for the robot with and without the use of SEA. A step further is taken in using all the actuated degrees of freedom of the robot to generate push recovery motions with the same approach, also considering the SEA. As there is a lack of literature and experiments of iCub walking, for this complex task we aimed at exploiting the capabilities of HeiCub by measuring its walking performances. We used the table cart model to generate walking trajectories on level ground, slope and stairs, which have never been achieved before by other iCub robots. In this way we could gain details of the platform that were unknown beforehand that are fundamental to be used in future optimal control formulations. Thanks to this study, future developments of walking control frameworks for the iCub family robots have now a point of reference.