Template:Nutshell AIFMD 21(3)

21(3): Eligibility criteria: a depositary must be

(a) An authorised EU credit institution;

(b) an EU investment firm meeting certain capital adequacy criteria;

(c) an entity subject to prudential regulation that is deemed ok under 2009/65/EC (EUR Lex)

For Non-EU AIFs the depositary can be an equivalent credit institution or investment firm outside the EU (I think). There are also exceptions for illiquid AIFs that don't invest in custody assets.