The overland map may contain multiple enemy forces, including lurking bands occupying hostile sites, more aggressive raiders infesting both civilized and wild lands, and open armies on the march making incursions.
Every known enemy force on the map is shown as a small coin on the map and also as a square monster portrait at the top right of the screen. (The coin may be hard to notice because it's often drawn overlapping with the heroes.) Scouting wild lands may reveal new enemy forces.
Clicking on either the coin or the portrait will bring up an assault job that can be used to initiate battle with that force.
Deepists are a minotaur-worshiping cult who live in cave systems. They put stone and fungi to creative use, and our heroes aren't sure whether their power comes from their own passion, or from something more sinister... Read some in-fiction lore here.
Drauven are grouchy lizard-folk who primarily live in the trees of forests and swamps, or in trees within swamps, but are happy to descend to the ground to go raid a village and maybe snack on a villager. They decorate themselves and their environment with bird and dragon-like motifs. Read some in-ficiton lore here.
Gorgons have been corrupting the local wildlife and turn them into raging tentacular monsters. They seem to be doing this experimentally, in pursuit of some deeper truth, but to call them "scientists" would tarnish the word. In the meantime, they continue to create the beasts as quickly as we can slaughter them. Read some in-fiction lore here.
Morthagi are clockwork undead, created from bone and metal long ago to serve some dark master. Their creator may be long gone, but the Morthagi themselves have persisted, clanking and hissing in the dusty, forgotten corners of the world. Read some in-fiction lore here.
All monsters are indeed monstrous. There may be shades of gray, but in aggregate monstrous life is incompatible with human life. Humans fight monsters to survive.
Every enemy force has a strength, which determines the number of foe cards it draws at the start of a battle. Strength starts at a base of 4 for incursion armies and 3 for other forces. It is increased by +1 for each territory the force has infested.
A "lurking threat" is a stationary enemy force occupying a hostile site.
Monsters are infesting this tile. They may ambush groups of three or fewer heroes. Patrol or take out the source to clear the infestation.
Every infestation has an origin force. The origin force may be an incursion army, or it may be a lurking force present at the beginning of the chapter. If the infestation has spread from an unexplored wild land, the force will not be revealed until you have scouted its origin territory. You can identify which infested territories belong to a given enemy force by hovering over either the force coin or the monster portrait, highlighting their territory.
Each territory infested by an enemy force adds +1 to its strength.
If infestations are not cleared away, they will spread to adjacent previously-safe tiles.
Infestations can be cleared either by patrolling or by assaulting the origin force. Patrolling first is not necessary, but it makes the battle easier.
Clearing away infestations earns more years of peace at the conclusion of a chapter, and allows the territory to be developed and produce resources.
The chance for a tile to become infested and the chance for small groups of heroes to be ambushed in them both depend on the difficulty level.
An incursion begins by spawning a monster army in a wilderness tile. They immediately begin moving into a developed territory -- this is shown as a red dotted line on the overland map. Hovering over the destination territory shows how soon the army will arrive. If desired, the monster army can be attacked in its origin territory right away.
When the army arrives, any heroes in the invaded territory must immediately fight a Defense battle. If they lose, retreat, or weren't there to fight off the army at all, the territory is conquered and converts from civilized back to wilderness. The monster army remains, and soon will begin moving again to invade another adjacent territory.
If the heroes fail to defend a town (lose, retreat, or are not able to fight), it will be destroyed upon occupation and can't be rebuilt.
Heroes can gain various advantages in the defense battle by performing the Prepare Defense job in the territory before the invading army arrives. Up to four different benefits can be gained by completing the job four times.
Many factors about incursions vary with difficulty: their frequency, their size, how fast they move, and how quickly they destroy constructions.