Sandbox Campaigns

Give the characters three problems at a time to deal with. Even if the characters solve one problem, the other two will develop and worsen, and possibly cause a new problem to arise. While the PCs might not be contacted directly to act against all three problems, make sure that they know about them through NPC chatter/concerns.

Example: An army of the undead is being lead by a powerful BBEG across the land towards some unknown destination. This has displaced a number of barbarian tribes who have become refugees and are forced into a kingdom's farmland. The kingdom sends the army to remove the barbarians. The disruption in the farmland has caused food shortages in the capital and riots/coup look likely.

If the undead aren't dealt with the BBEG acquires a location/artifact/macguffin of power and further damages the land/displaces more threats to the kingdom.
If the barbarians aren't dealt with they engage the army in battle making the farmland even more dangerous.
If the food shortage isn't dealt with riots start with hints at a coup becoming a serious concern.

So three themes need to be constant:
A BBEG threat, which you should always be dropping the PC's clues and hints about, whether they be rumours or items.
A Regional threat, which is large scale,but not directly on the PC's unless they get involved with higher powers.
A Local threat, which changes from scene to scene and may worsen if they leave or do nothing.
The threat does not have to be towards the PC's, but if you can form a tangent that makes them notice how it affects them, then perfect

And don't forget to throw side quests out for specific PCs to pursue especially if it relates to their backstory.
Maybe the barbarian refugees are of a rival/same tribe as one of the PCs.
Maybe it's unknown why the barbarians entered the farmland in the first place since one PC's beloved relative forged a truce with them.
Maybe the leader of the rioters is a friend of one of the PCs.
Maybe there is loot left in the wake of the undead army and hopefully one PC's former bandit friends haven't moved in yet.
Maybe one PC's religious/knightly order is fighting a stalling battle to slow the undead so that the peasants can escape.

Basically make sure the world is always moving and the PCs never feel like they are the focus of the world. That is until they get to be of such power and reputation that the fate of the world does indeed rest on their shoulders.

At which point it's time to introduce the Bigger Fish.
Unlike IRL, in DnD and similar if you get into great power, the power-that-be, both good and evil, will approach the characters with ancient contracts, faith-shattering information, and in general just makes the world suddenly appear a whole lot bigger, because to all the Infernals, Celestial and otherwise, the Prime Material is their battleground, and you are their pawn.

Lastly, an excellent little summary I read recently on /tg/:
Remember three names: Chekhov, Schrodinger, and Samus.
Schrodinger: Everything exists in a state of superposition, so that at least some specific details are not set in stone until the PCs observe and act upon it.
Chekhov: Anything given meaningful detail should be significant.
Samus: If the PCs can't possibly get there now, there should be some way of giving them the opportunity to later on.