The premise of an Academy is that you can only have 2 Academies in any one city ... and my "own" additional limit is that you can only have 4 total academies (small/medium maps) or 6 total academies (large/huge maps) total.

Each Academy is a fairly significant investment, taking up 4 spaces "one game tile"

The ability to build a Diplomatic Academy is early on in the Diplomatic tree (tier 5 or so), War Academy is in the Warfare tree, Adventurer's guild is in the Adventuring tree, Library of the Arcane is in the Magic tree, and everything else is in the Civics tree.

The Diplo, War, Magic, and Adventure academies are all unlocked relatively early within their own specialties (tier 4 or 5) ... while all the other academies are late Civics tech, on par with the tech for Universities, or around those parts.

There is no limit to the number of any one particular academy type you build ... so if for instance, your on a medium map, with only 2 cities, you can build two adventurer's guilds in each city and have those for the whole game.

Diplomatic Academy

• +3 research
• +2 research, for every ally you have now (if researching Diplomacy tech)
• +1 research, for every war you have ended in a diplomacy screen (if researching Diplomacy tech)
• -1 research, for every war you have started (if researching Diplomacy tech)
  
• +1 research for every non-ally you have an agreement/ treaty with (if researching Diplomacy tech)
• +3 skill for any diplomats trained here(?)

Scholars Academy (generic)

• +4 research
• +1 research, for each empire you are trading with
• +5 research, if your empire is the least technologically advanced in the world and you have contact with at least one nation more advanced than you
• +1 research for every research producing unit stationed in the city (scholars)

Adventurer's Guild

• +2 research
• +2 research for each legendary monster you've slain
• +1 research for each legendary artifact you own
• +1 research for (1x quests completed/5)
• each time you complete a quest, adds +10 research for that turn only. Adds less research for weaker quests, and more for complex quests, always a one time bonus.
• Increased chance for rare artifacts to show up in the city's shop
• adds additional Champion training for costs of Gold (only so many trainings for each Champ?)

War Academy

• +3 research
• + 5 research (if researching Warfare)
• + (1 * Net battles won) research (if researching Warfare)
• + (1 * enemy soldiers killed/ your own soldiers killed) research (if researching Warfare)
• +1 research for each Military building present (if researching Warfare)
• -50% troop training time

Library of the Arcane

• +3 research
• +4 research (if researching Magic)
• +1 research for each spell learned (if researching Magic)
• +(8 + Intelligence/ 4) research if Sovereign is present (if researching Magic)
• all magical buildings in city produce +1 spell points (cumulative)

Civil Service Academy

• +3 research
• +4 research (if researching Civics techs)
• +2 research (if researching Diplomacy techs)
• +1 research for every 20 turns you have not declared war, attacked an opponents settlement, or lost a city via conquest (if researching civics or diplomacy)
• -40% maintenance

Majestic Royal Academy

• +3 research
• +(3 + Charisma/2) research if Sovereign is present in city (if researching Civics or Diplomacy)
• +2 research for every family member present in city (if researching Civics or Diplomacy)
• +4 gold
• +1 gold and research for every level 5 settlement (foreign or domestic) connected by caravan by ONE degree (meaning a direct "friend" not a "friend of a friend")
• +1 prestige for each government building (palace, town hall)

Merchants Academy

• +2 research
  
• +1 gold for every caravan that arrives from a domestic city
• +3 gold for every caravan that arrives from a foreign city
• +1 gold for every gold producing building
• +1 research for every trade treaty
• +1 research for every foreign trade route (in city)
• items in the shop are cheaper

Academy of the People

• +3 research
• +(1 * population/ 100) research (if researching Civics tech)
• +1 prestige * city level
• -(1 * soldiers in city/ (population/10)) research (if researching Civics tech)
• +1 research for non-royal champions already producing research staying in the city