In the Bible dictionary, under Devil, it says the following: "Since the devil and his premortal angels have no physical body of flesh and bones, they often seek to possess the bodies of mortal beings. There are many such instances recorded in scripture (Matt. 9:32; 12:22; Mark 1:24; 5:7; Luke 8:30; Acts 19:15; see also Mosiah 3:6). Such can be evicted by the power of faith in Jesus Christ and the exercise of the holy priesthood. Jesus gave this power to His disciples (Matt. 10:1; Mark 16:17; Luke 10:17; Acts 5:16; D&C 84:67)."
We know that his influence is everywhere, deceiving and leading people away from God but I have pondered over these scriptures of Christ and His disciples casting out devils and wonder, do we still see or hear about accounts of people being possessed by devils? I haven't. Although I can't say I read the news or watch much in the way of documentaries or anything.
Perhaps this is a subject that is too uncomfortable. I know that when speaking of the Holy Spirit it brings peace. Quite the opposite is true when speaking of the adversary. Perhaps it's one of those things that is seen/felt, prayed about, cast out, and moved away from. I don't think we are to dwell or ponder greatly on the things of the devil, rather those things that are from God. I personally believe that possession still happens though I don't know anything about the specifics or who it might happen to but I do know that the Holy Spirit works in us to identify what is needed and through His guiding influence, the Priesthood Power and our obedience, the devil will have no power over us. I feel safe and comfortable in that.