The 'Great Man Theory' is one that was popular in the early-mid nineteenth century. It suggested that "the course of human history results largely from the actions of Great Men", or in other words, that certain figures were responsible for the bulk of the goings on in the world.

An example would be Napoleon Bonaparte. Rising to the heights that he did, he plunged Europe into war, and the events that followed his campaigns undoubtedly had an effect on Europe in the century that followed.

The counter-argument to this theory came via Herbert Spencer, who said:

Those who regard the histories of societies as the histories of their great men, and think that these great men shape the fates of their societies, overlook the truth that such great men are the products of their societies.

Basically, Spencer is claiming that great men wouldn't necessarily exist outside their societies. If you were to remove Napoleon from eighteenth century France, where he grew up, and instead placed him in another place and era, his life would take a different path, not guaranteed to be one of 'greatness'. Spencer's counter-argument asks us to inspect the context of these 'great men', rather than just accepting their greatness as inherent.

This is in effect, one of the cons of 'great man' history. We are lulled into thinking of certain men as the key figures of a time or place, and neglect to look beyond to what made them that way. We are also led to believe that these people are somehow the sole forces for change or moves of history. If history views events this way, then it forgets the people who participated. Napoleon invaded Russia with over half a million troops. A great man history reduces those men to cannon fodder, who are of little or no use to the historical record. Yet if studied, we can learn about society at that point in time - the feelings of the soldiers towards the war and their officers, how their lives were affected by war, and any number of other bits of information that may come up in journals, diaries, and letters.

From my area of study, the most obvious choice would be Martin Luther King, who is the person of the Civil Rights movement. He was a brilliant man, an eloquent speaker, and he oozed charisma. But while he was essentially the frontman of the movement, he wasn't the reason it happened. He couldn't have achieved all that he did without the help of SNCC, and the SCLC, and the NAACP. Before MLK was born there was half a century of black(and white) men and women fighting for equality in America. But the position of MLK at the time of the CRMs greatest victories meant that he is remembered as the cause. The cost is that the churchwomen, and educators, and student groups, and railway men, who all organised activism across the South are often forgotten by popular history. EDIT: As /u/TheShowIsNotTheShow pointed out in his reply, the greatest downside is the erasure of everyone except 'the man'. That's essentially what I wanted to say here, but it wasn't very clear.

And this is where I would tenuously suggest that the positive side of 'great men' history is that it entices people to history through the actions of these popularised figures. People get interested because they hear names like Henry the Eighth, or Charlemagne, or your example of Genghis Khan. Not as many people will be drawn to history by John Smith, the baker who baked a cinnamon bun that the Prince of Avignon choked on in the 16th century. But tell that same person about Mansu Musa, the Malian King who destroyed the economy of the eastern Mediterranean by giving out bags of gold dust to Egyptian beggars, and they might be lured into asking more questions.

So the upside of 'great men' history is that it creates these leading roles, and they act as lighthouses for this field, guiding people in to a certain area of history that intrigues. The hope is that once you've reached that safe harbour and begin to learn about that history, you'll dig deeper and explore the society that these men come from, and the context which led them to become conquerors or despots, heroes or villains. Or both?

Source: My main source is this article - (PDF warning) about Herbert Spencer, the man who countered the 'Great Man Theory' with his argument that great men are formed by their societies. The rest would just be knowledge gleaned through my studies, and I can try find some more articles if anyone wants them, although I suspect that they might be paywalled.

Great Man history is the notion that individual actors are the engines of history. Things happen because people like Alexander, Caesar, Napoleon, Hitler, etc. make things happen. It ignores economics and other factors that explain how such people got into power and why their ideas had an effect on the world.

The disadvantage is that whatever decisions these people made, most likely someone else could have made them also, and the vast majority (the poor) probably didn't care. New boss, same as the old boss.

Great Man theory also doesn't present a systematic theory of history. Special people just pop up; there isn't a longue durée. Each case has to be examined independently, because Great Men are the exception (most of us are not great).

I can't think of an advantage to GM theory. Maybe it's just too late and my mind isn't working anymore.

The obvious pro to great man history is that it tells us much more about the individual than other forms of history. The example that I always use is that of Beethoven. If he had not written the music that he did at the right time then the Romantic era would never have begun in the way it did. His music and his musical language is hugely important to musicology. The flip side to that is that focusing on Beethoven alone and not huge changes of his era leaves a huge amount out that cannot be connected to one individual. For example, Beethoven was a jobbing composer, and was not attached to a royal court for a job and patronage. He was able to do this because of a new system of music publishing which meant that he had enough income to survive, the growth of the middle classes who played his less difficult works and songs in their living rooms and a horde of other socio-economic factors.

The individual of course has a huge role in history, but examining history solely from the point of view of the individual leaves far too much out.