Barracks: In general, I say build it. When is a bit complicated, and depends on other strategy decisions you've made. Generally I like to build them before I train the Trebuchets I will turn into Canons, so that they start with at least one promotion. You could build them earlier if you have enough troops for the moment and it won't take too many turns. The biggest bonus they give is if you get to ideologies and choose Autocracy.



Circus: All happiness buildings should be built when you need them. Since Circus requires the least hammers, I usually build these pretty early. Depends entirely on how much happiness you have.



Granary: A key building for all cities to build. When it gets built depends on your social policy tree. See the specific strategies below for more information on build order. But you will want them, so that you can grow your cities a bit to reach the military techs you need a bit faster. I've played games where I didn't build them at all and it made up to 20 turns difference on reaching Dynamite. Whether that is inevitable I'm not sure, perhaps other factors were involved. But these are inarguably important buildings.



Library: You want these as soon as you can. For most of my games, I set my expos to build: Monument, Archer, Library. I'm not a fan of rush-buying Libraries like the peace mongers advise. I'd prefer to prioritise workers on forest chops to get them faster. You should be stealing as many workers as you can anyway. If you don't like stealing workers, don't place Domination. It is pointless to throw away potential advantages because you think worker stealing is an exploit when what you aim to do (militarily defeat the AI) amounts to being an exploit when judged on the same criteria.



Monument: If you go Tradition, you will get these for free. If not, build them after a scout in your capital, and as the first thing in your expos. Pretty obvious why. I can win Domination games without granaries but its not so easy without Monuments. They will probably be the only culture buildings you build, so you need them, to reduce the bottleneck you will experience before you capture a couple of culture-heavy capitals.



Shrine: There is no consensus on this, but I'm personally of the opinion that it's not worth founding your own religion when going Domination. When you play peaceful, not emphasising religion means your FPT will be low throughout the game. However, when you've captured a few AI cities, your FPT will be high. The only times I ever build a shrine is when I have a decent food pantheon and I'm in a position where the pantheon is NOT gonna get knocked out before T150. That's my view, and although I'm sure this could be endlessly debated, it doesn't really matter so much. I don't like to waste the hammers unless I'm getting food (=growth). If someone can make a conclusive case for bothering with shrines in a DomV on Immortal/Deity, then I'd love to hear it.



Stone Works: Only if you have 2 or more sources of stone, and/or need the happiness.



Walls: Only later in the game to get happiness from Autocracy. You are the aggressor. You don't need defence.



Water Mill: Only after the 1st army has departed on its warpath and only if there is nothing better to do. You don't need the small boost to growth or production early.



Ampitheater: A waste of hammers in most cases. Again, like the shrine, if someone has a case to make, make it.



Aqueduct: Build these as soon as you can. More growth = faster military techs.



Caravansary: Generally a waste of time.



Colosseum: When you need it.



Lighthouse: In a coastal city if you have sea resources and/or are food starved.



Market: Very important for Domination. Keep your economy in good shape. I usually build these ASAP in expos, after Monument, Library and a couple of units.



This takes us up to the Medieval era. Since your army should be built and abroad by now, you can build more buildings hereafter, but just remember the golden rule:

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