He has quite the interesting life story. Ascending to the throne at age five, he was initially sidelined by his country's fascist government. By the time he turned 22 he was old enough to know better and backed a coup to depose Romania's infamous pro-Nazi prime minister, Ion Antonescu, installing a pro-Allied government in his place. For this the Western Allies rewarded Michael by handing his country over to Stalin, who in turn installed a Communist regime that ended the monarchy.
Today the 85-year-old Michael lives in Switzerland and occasionally goes back to Romania for brief visits, now that the commies are gone. He hopes he'll be restored one day, but that seems rather unlikely.
Michael is unusual among living deposed monarchs in that he is actually a fairly popular individual who people generally respect. Not enough to go through the hassle of restoring the monarchy, but a quiet reverence all the same.
There are a handful of other deposed monarchs kicking around the world, but they are all generally controversial figures, if not outright unpleasant and actively disliked.
King Constantine the Second of Greece (born 1940) gave royal approval to a military coup in 1967 that put an end to a series of faltering interim governments and installed a right-wing dictatorship. A few months later he felt guilty and tried to promote a counter-coup, but the generals quickly put that down and Constantine was forced into exile. The Greek people never forgave the King for his incompetent meddling, and when democracy was restored they voted overwhelmingly to permanently make Greece into a republic. He now lives in London.
Czar Simon the Second of Bulgaria (born 1937) has a life story that is sort of similar to Michael's. He became Czar in 1943, at age six, after his father was poisoned by the Nazis. Bulgaria tried to stay neutral in the war but the country was eventually invaded by the USSR and the monarchy was abolished in favor of a pro-Stalin puppet regime. Exiled, Simon was raised in Spain and dreamed of one day becoming a politician. When the commies left, he returned to Bulgaria and fulfilled his dream, founding a political party. He was elected prime minister in 2001, but then lost his bid for re-election in 2005.
Mohammed Zahir Shah of Afghanistan (born 1914) became king at age 19 after his father was assassinated in 1933. He repeatedly clashed with the government of the hardline prime minister Mohammed Daoud Khan, and in 1973 Khan changed the country into a republic so he would no longer have to share power with the monarch. The King moved to Italy where he lived for almost 30 years. Then in 2002 when the Taliban were deposed he returned to the country with American backing. Some Afghanis regard him with nostalgia, as a symbol of a more peaceful time. Others think he's just some doddering old geezer. Regardless, Afghanistan is too much of a tribally-divided society to consider restoring a monarchy, as that would implicitly entail placing one tribal group symbolically "above" the others.
I think there are a number of factors explaining why you don't see monarchies abolished as much these days as you used to:
1- All western monarchical countries (ie: Europe and Japan) have democratized their systems of government to the point where the king or queen is entirely powerless, and thus politically uncontroversial. Most of the European monarchs who were deposed in the 20th Century were singled out because of their unwanted meddling in the affairs of the civilian government.
2- There is no real precedent for how to "politely" remove a monarchy. As mentioned, most monarchs deposed in the last century were kicked out on bad terms, with strict laws being passed shortly thereafter that banned the royal family from ever setting foot in their former realms again. If, say, Britain was to become a republic tomorrow, it would seem excessively cruel for the British government to order the royals out of the country. After all, they themselves have done nothing wrong... they're just part of a bad institution. But if they stayed, the legitimacy of the new republican government would be undermined, since presumably the media and tabloids and general public would still focus on the affairs of the princes and princesses- even if they no longer legally held such titles. So what can you do, really? Many socialist European politicians find this dilemma to be more of a hassle than it's worth. This is why, for example, Sweden still has a monarchy.
3- Outside of the west (ie: the Middle East) monarchs who are in currently power are very authoritarian leaders who have successfully established a regime of oppression to quickly crush any opposition to their rule. The monarchy is the government, not merely a "part" of it.
In short, to survive a monarchy has to be either entirely democratic (and thus powerless) or entirely authoritarian. Monarchies have to pick one of the two choices and evolve permanently in that direction. A monarchy that tries to be sort of authoritarian and sort of democratic at the same time is a monarchy that is bound to fail. This is what our friend in Nepal has recently learned.
Then of course, there is the distinct debate over Commonwealth monarchies, like Canada, Australia, Jamaica, etc. But that's for another time.
The surviving monarchs of Europe.