All 3 factions has a bank within the first major city (where you spawn after the initial quest) and normally have 2 or so per zone. The bank itself has an icon which looks like this (unless of course this is my addon sort of thing and I've forgotten how the basic game layout is, as it has been quite a while):
Edited by The_Sadist on June 12, 2015 4:25PM
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Note, there are only one or two cities in each zone that have a Bank, so keep that in mind as you play on. Most zones only have one city that have a bank.
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It has bank name and city locations in it. Aldmeri Dominion Mistral Treasury - Mistral, Khenarthi's Roost, Elsweyr Vulkhel Guard Manor & Treasury - Vulkhel Guard, Auridon, Summerset Isles Skywatch Coinhouse - Skywatch, Auridon, Summerset Isles
Daggerfall Covenant Dugronk's Impenetrable Fort - Port Hunding, Stros M'Kai, Hammerfell Bank of Daggerfall - Daggerfall, Glenumbra, High Rock Bank of Betnikh - Stonetooth Fortress, Betnikh, High Rock Wayrest Treasury - Wayrest, Stormhaven, High Rock Shornhelm Savings & Loan - Shornhelm , Rivenspire, High Rock Bank of Sentinel - Sentinel, Alik'r Desert, Hammerfell Bank of Evermore - Evermore, Bangkorai, High Rock
Ebonheart Pact The Bleak Vault - Bleakrock Village, Bleakrock Isle, Skyrim Uveran Bank - Davon's Watch, Stonefalls, Morrowind Drinks-With-Toes - Dhalmora, Bal Foyen, Morrowind Windhelm Bank - Windhelm, Eastmarch, Skyrim