Well, in Japan, from the Heian period to around the time when the Mongol invasions happened, the very front of the army was considered a place of honor and the most loyal and trusted vassals would be given the privilege of standing at the front, likely with as many horses and attendants/guards as possible.

It was a symbol of status.

Of course, the reason why this was a privilege and not a death sentence for Heian period warriors was because most of the battles during this time were not the brutal, dirtier affairs that were Sengoku period battles.

This was also the time period when the samurai class was first beginning to form.

This was the first class of people's in Japan that identified as a professional military, their martial skill, discipline, training, and loyalty being their main assets.

Before this, anyone who wanted to rise in status would have distanced themselves from something as uncouth as violence. Part of this has to do with Shinto but I digress.

During this time period, when professional soldiers met in battle, it would have been very ritualistic and would have mostly taken place from afar, with exchanges of arrows from horseback being the main form of combat.

Close quarters combat becoming the norm would only occur after the Mongol invasions, during the many battles of massive levied armies during the Sengoku Period. The famous samurai sword would have mostly been a last resort or, more likely, used when taking the heads of their dead opponents to bring back. These heads were important for promotions and laying claims to kills.

During this time, the mounted warriors armed primarily with bows became the main military force of major political players. In this context, the front line was not so much a safety risk as it was a place of status and glory.

This started to change with the Mongol invasions. The invaders did not wait for names to be called, challengers to ride out. They did not demand single combat for clear kills. They came on in waves, attacked without warning and did not care about the protocols. This changed Japanese military doctrine a great deal, as one can imagine.

During the Sengoku period, this changed even further. With the press of thousands upon thousands of infantry men who were deadly with spears, bows, and firearms, the front line was no longer safe. But that didn't stop the symbol of the front line from still being associated with glory.

When arrayed on the battlefield, those with the highest status were still accorded their privilege of standing on the front line. But as battle was joined, they would pragmatically move out of the way for the massive numbers of infantry to do their bloody work while they waited for an opportunity to strike.

In actual battle during the Sengoku period, the front line soldier would depend a great deal on the formation that the general picked at that moment. Generally, one could expect arquebusiers to be part of the front line, since they needed direct line of fire to reach their enemies.

And because of the nature of the arquebus, this naturally meant that often mounted samurai were not on the front line when battle was joined. That being said, there were some who only allowed samurai and men of status handle expensive weapons like the arquebus.

So in Japan, the answer to your question varies a great deal depending on the time, place, leader, context of the battle, etc. I think this is true of most any place, really.

For example in China, depending on the tactics for the battle, the elites might have been placed in the center front and flanks, with the largest men they could find standing on the front line to be intimidating. Other times, it might have been mounted soldiers, who would have been by necessity wealthier than the infantry behind them.

In general? There's pros and cons to having your best men up front and center. There's added risk to being in front, sure but it wasn't an immediate death sentence just by standing there.

You'll often see in movies or games that depict pre-modern melee combat as a meat grinder, with the men in the front dying within seconds of contact with the enemy, often times seemingly dying through sheer momentum and unavoidable contact with dozens of enemies at once.

If this devastating death by charge doesn't take the men first, then they'll quickly fall to enemies only to be replaced by another soldier in the second line, who is now thrust to the deadly front.

This is not how most battles would have been conducted, especially if the men in the front have anything to say about it. For certain, most men are not all that willing to charge straight to their deaths for no certain gain simply on orders. Actors are a special breed of warrior.

If you have more specific questions or just follow up questions, I'd be happy to answer more.

Goldsworthy, quoting Xenophon (Greek) and Frontinus (Roman) says that the bravest men were kept in the front and back. The front guys need to be brave to actually fight, while the guys in the back need to be brave to not only not run away but also keep other people in formation from running away. (IIRC at Hastings the Housecarls were placed in front)
The depth of the formation depends on both weight necessity and skill of the troops. Greener troops have lower morale and less ability to maneuver, so they are kept in deeper formations, while better troops who are braver and can better maneuver are kept in shallower formations. A formation seems to never be fewer ranks than three, and often four or more.

The Roman Army At War - Adrian Goldsworthy

Edit: Almost forgot. The best troops were in general kept on the right because soldiers were protected in part by shield of his comrade to his the right, and the soldier on the extreme right is obviously deprived of this protection.