The abundance of X-ray binaries in our and other Galaxies and their brightness at many wavelengths make them key to our understanding of different physical processes (accretion and gravitational waves radiation) and of advanced questions in stellar evolution (the origin and evolution of binary systems, the final stages of massive stars, the formation of compact objects). Within the next decade, observations at existing and planned facilities in almost all wavelengths will put a heavy demand on further studies of compact binaries. In my talk, I will discuss what is our current knowledge of low-mass X-ray binaries formation, where we had a progress and where we have failed, what are new breakthrough ideas.
Specific attention will be given to globular clusters, where dynamical interactions between objects lead to the enhanced formation of X-ray binaries. The only detected so far black hole binary with a white dwarf companion is located in a globular cluster, and its existence gives us the key what the unseen population of black holes in globular clusters could be.