The review addresses the central concept of the uncontrolled manifold (UCM) hypothesis, which has become a major framework for analysis of performance-stabilizing motor synergies. The major goals are to summarize the status quo in the field and to ask new questions stimulating new studies. We focus on a few main questions: What is the UCM? What are the likely neural origins of the UCM? How is the UCM reflected in movement patterns? Are properties of the UCM similar in all directions? We contrast experience-based features of movements seen very soon after the movement initiation vs. those based on on-line sensory feedback signals. Further, we address a number of poorly explored issues such as the differences in characteristic times of processes within the UCM and orthogonal to the UCM space, the interplay between movement stability and optimality, the origin of preferred sharing patterns of performance variables across abundant sets of elements and of their inter-trial variability, problems with the UCM-based analysis in different spaces, and likely neurophysiological mechanisms contributing to the UCM formation. In particular, we focus on the UCM in spaces of hypothetical neural control variables, which we associate with the reciprocal and coactivation commands to the effectors. Analysis of performance-stabilizing synergies within the UCM framework in abundant spaces of kinetic, kinematic and electromyographic variables at the selected level of analysis may be practically useful. However, mapping findings in such studies onto neural control mechanisms has been challenging.