In the prosthetics field, one of the most important bottlenecks is still the human-machine interface, namely the socket. Indeed, a large number of amputees still rejects prostheses or points out a low satisfaction level, due to a sub-optimal interaction between the socket and the residual limb tissues. The aim of this paper is to describe the main parameters (displacements, stress, volume fluctuations and temperature) affecting the stump-socket interface and reducing the comfort/stability of limb prostheses. In this review, a classification of the different socket types proposed in the literature is reported, together with an analysis of advantages and disadvantages of the different solutions, from multiple viewpoints. The paper then describes the technological solutions available to face an altered distribution of stresses on the residual limb tissues, volume fluctuations affecting the stump overtime and temperature variations affecting the residual tissues within the socket. The open challenges in this research field are highlighted and the possible future routes are discussed, towards the ambitious objective of achieving an advanced socket able to self-adapt in real-time to the complex interplay of factors affecting the stump, during both static and dynamic tasks.