In young star clusters, the density can be high enough and the velocity dispersion low enough for stars to collide and merge with a significant probability. This has been suggested as a possible way to build up the high-mass portion of the stellar ma
ss function and as a mechanism leading to the formation of one or two very massive stars (M > 150 Msun) through a collisional runaway. I quickly review the standard theory of stellar collisions, covering both the stellar dynamics of dense clusters and the hydrodynamics of encounters between stars. The conditions for collisions to take place at a significant rate are relatively well understood for idealised spherical cluster models without initial mass segregation, devoid of gas and composed of main-sequence (MS) stars. In this simplified situation, 2-body relaxation drives core collapse through mass segregation and a collisional phase ensues if the core collapse time is shorter than the MS lifetime of the most massive stars initially present. The outcome of this phase is still highly uncertain. A more realistic situation is that of a cluster still containing large amounts of interstellar gas from which stars are accreting. As stellar masses increase, the central regions of the cluster contracts. This little-explored mechanism can potentially lead to very high stellar densities but it is likely that, except for very rich systems, the contraction is halted by few-body interactions before collisions set in. A complete picture, combining both scenarios, will need to address many uncertainties, including the role of cluster sub-structure, the dynamical effect of interstellar gas, non-MS stars and the structure and evolution of merged stars.
Massive black holes (MBHs) with a mass below ~ 1e7 Msun are likely to reside at the centre of dense stellar nuclei shaped by 2-body relaxation, close interactions with the MBH and direct collisions. In this contribution, we stress the role of mass se
gregation of stellar-mass black holes into the innermost tenths of a parsec and point to the importance of hydrodynamical collisions between stars. At the Galactic centre, collisions must affect giant stars and some of the S-stars.
