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.
Previous Owners include: Heimdall, Loki, S.H.I.E.L.D., Howard Stark, Hydra, "Church Keeper"
##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.
Previous owners include: Ronan, Star-Lord and the GotG...
##Time Stone is owned by Doctor Strange
Doctor Strange owned the gem as part of the Eye of Agamotto, the stone was used by Doctor Strange, first to aid him in his learning of sorcery, then in his final fight against Kaecilius and Dormammu. It is unknown how long the Stone has stood as part of the Eye on a pedestal in Kamar-Taj.
Previous owners include: Calgiostro, Agamotto
##Soul Stone is Unknown
The location of the Soul Stone is revealed for the first time in Avengers: Infinity War