The most famous disappearance that comes to my mind is Jimmy Hoffa. He was a prominent trade union leader in the 1950s and 1960s. On July 30, 1975 he went missing and was never seen again, nor was his body ever found.
Maybe this is more of an opinion question, but is he the most well known person to vanish off the face of the earth in recent history?
This website lists 10 celebrities who disappeared and have never been found; while several of them are more recent than Hoffa, none of them were anywhere near Hoffa’s level of prominence, at least in the U.S. The ones which I recognize are Bison Dele (former basketball player for the Chicago Bulls) and Scott Smith (former bassist for the rock band Loverboy).
Amelia Earhart was pretty well known at the time of her disappearance.
ETA: The previous Ninja link includes her.
In the UK, possibly Lord Lucan. Or possibly the Australian Prime Minister Harold Holt.
Indeed, and I didn’t mention her because I interpreted the OP as asking about disappearances which were more recent than Hoffa’s.
This may not be what you’re looking for, but Agatha Christie disappeared for 11 days in 1926. It was a big news story at the time, and what happened is still a bit of a mystery.
The thing with Hoffa, though, is that there is little doubt that he was killed. It’s just his body that has “disappeared”. The ultimate fate of Earhart or some prominent Nazis* is/was more of a mystery.
*We often find out the fate of fugitives like that after they have died.
It looks like Ambrose Bierce - author of The Devil’s Dictionary - disappeared, after planning a trip to Mexico.
Famous to me, at least. It’s a good book.
Sorry, I didn’t mean to restrict it to events more recent than Hoffa’s.
I think it would make sense to limit it to the last few centuries though.
I saw what you did there. 
Given the OP’s clarification on “recent history,” it seems like Earhart and Hoffa are likely at the top of the list, as far as the most prominent to have disappeared and never been found/recovered. Holt and Lord Lucan (and their disappearances) might well be better-known in their home countries than Hoffa’s is.
Agreed - cases where suicide or murder are obviously plausible explanations, where it’s really just the absence of a body, are less compelling. Unless it’s hard to see how the body could have disappeared.
True, Hoffa doesn’t mean much to me, and the question brought to mind Lord Lucan immediately. Although I suspect Lord Lucan is mostly famous because he went missing (after murdering the nanny), than for being a famous person who went missing, if you see what I mean.
Of course, Lord Lucan’s disappearance is also tantilising because there has been endless speculation about where he went - did he drown himself or is he still living on a ranch in South Africa?
Interesting point. 
I was a kid when Hoffa disappeared, though I suspect that, as one of the U.S.'s most prominent labor leaders, in an era in which unions had quite a lot of power, he was reasonably well-known among the general public.
But, when he disappeared, I remember it being all over the nightly TV news for weeks. And, like Earhart, there is still an awful lot of speculation about what happened to him, and there’ve been many TV shows which explore the mystery. When a rumor started, many years after the disappearance, that Hoffa might have been buried underneath one of the end zones at the then-under-construction Giants Stadium in New Jersey, it became a running joke for years.
The Master spoke: Why did mystery writer Agatha Christie mysteriously disappear?
Bandleader Glenn Miller in 1944
Aimee Semple McPherson was awfully famous in her time.
Is Harold Holt really a mystery, though? The man decided to take a swim off a coast known for strong and erratic currents. He presumably got in trouble and drowned. As Bill Bryson put it in In A Sunburned Land - which is where I heard about it - the only thing unusual about the story was that Holt was Prime Minister of Australia at the time; if he were just some schlub, there would have been no mystery at all.
Bryson also pointed out that Australians erected a wonderfully Aussie memorial to Holt - a municipal swimming center.
Wallace Fard: the movement he founded (Nation of Islam) is a bit more famous than he is, but he’s an interesting character with an interesting backstory and an unsolved disappearance.
Not very mysterious–that wikipedia article acknowledges some crackpot CTs, but basically says his plane got iced up and crashed. It was a disappearance in that his body was never found, but otherwise it seems pretty straightforward.