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- "Blast - Build - Battle"
– Motto of the U.S. 16th Engineer Brigade
- "The more science intervenes in warfare, the more will be the need for engineers in the field armies; in the late war there were never enough sappers at any time."
– Bernard Montgomery
Military Engineering is a Medieval Era technology in Civilization VI. It can be hurried by building an Aqueduct.
Strategy
With engineering all the rage in the Middle Ages, it was only a matter of time until it got a specialized military application branch. The development of weapons and countermeasures led to the need to develop countermeasures to the countermeasures, and so on. And here it is where it became obvious that a special corps of half-scientists, half-builders would be needed on the battlefield. Enter the Military Engineer - the best friend of a wise general.
But it didn't stop there, of course. The Encampment gained a new building, specifically concentrated on providing technical support for military purposes: the Armory. Finally, the first Resource of the next-generation military tech was revealed: 
Niter.
Military Engineering is an important tech overall, because of its revealing a new Strategic resource (which has very nice yields, even if you don't consider its military applications). But militaristic civs will want it quickly anyway, because of the Armory building it unlocks, which will further boost your army's rate of acquiring Promotions. Plus, it leads to Gunpowder, which provides the next-gen frontline unit, the Musketman. To beeline for it, you need to research the central part of the tech tree, which usually happens naturally; then from Construction go straight to Military Engineering to get an instant economic boost.
Civilopedia entry
Loosely defined as “the art and practice of designing and building military works and maintaining lines of military transport and communications,” military engineering dates back to the Roman legions, which each had a small, specialized corps devoted to overseeing the building of fortifications and roads. They were also the ones to build the catapults, battering rams and siege towers when needed to stamp out some unruly town. But for over five centuries after the fall of Rome in the west military engineering barely progressed; it wasn't until late in the Middle Ages that the need for siege warfare again spurred the advance of military engineering.
With the development of gunpowder, military engineers became vital, both in designing fortifications to withstand cannon (one of the first innovations: earthen walls worked better than stone ones, since the cannonballs just went thunk and sank into the dirt) and devising ways to get the cannon close enough to the fortifications to be effective. Military engineers became specialized. Sappers, for instance, first appeared in the French army; their task was to excavate zig-zag trenches towards the enemy walls to protect infantry and artillery, all while under musket and cannon fire. In the British army, miners tunneled under enemy walls to place explosive charges ... not a long-term career.
Meanwhile, Renaissance engineers were devising new types of forts and weapons; even da Vinci got into the act with his flights of military fancy. More practical engineers, such as French Marshal Sébastien Le Prestre de Vauban (1633–1707), developed new types of forts and new tactics to use against them during the wars of Louis XIV. During the brief times of peace, military engineers turned their attentions to fortifying the borders and building and maintaining roads and bridges to facilitate the movement of troops. With the coming of the telegraph in the 1800s, they also got stuck with maintaining the lines of communication and command. By the time of the First World War, military engineers were indispensable in modern warfare.
