Work in Progress
The Infinity stones have changed hands various times over the history of the MCU, some more than others, these are the current owners:
##Space Stone is owned by Loki
This is confirmed in this Q&A as Loki having stolen it from the vaults in Asgard before Surtur destroyed Asgard.
##Mind Stone is owned by Vision
The Avengers claims the Mind Stone from Ultron when they steal the cradle with Ultron's new body in it. Thor, Stark and Bruce Banner grant the body life creating Vision.
Previous owners include: Ultron, Avengers, Hydra, S.H.I.E.L.D...
##Reality Stone is owned by Taneleer Tivan (The Collector)
The Asgardians gave the Aether (and the Reality Stone) to The Collector for safe-keeping after defeating Malekith.
Previous owners include: Asgard, Malekith, Jane Foster...
##Power Stone is owned by Nova Corps
The Power Stone is given to the Nova Corps for safe-keeping after Peter Quill catches the unprotected Stone and destroys Ronan the Accuser.
##Time Stone is owned by Doctor Strange
##Soul Stone is Unknown