AnyObject and Any in Swift

AnyObject is only for reference types (classes), Any is for both value and reference types.

So you should go for [String: Any].

Type Casting for Any and AnyObject

Swift provides two special types for working with nonspecific types:

• Any can represent an instance of any type at all, including function
  types.
• AnyObject can represent an instance of any class type.

NOTE:

Use Any and AnyObject only when you explicitly need the behavior and
capabilities they provide. It is always better to be specific about
the types you expect to work with in your code.

From The Swift Programming Language:
https://developer.apple.com/library/content/documentation/Swift/Conceptual/Swift_Programming_Language/TypeCasting.html#//apple_ref/doc/uid/TP40014097-CH22-ID342

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Also note that when you work with Cocoa API, it’s common to receive an Array of AnyObject, this is because Objective-C arrays are NOT typified.
So you need to cast them to the array type you expect.

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EDIT: (december 22, 2015)
On the last statement, note that this is changing with Swift 2.0 and Xcode 7.
Apple has introduced ‘Lightweight’ generics in Objective-C so lots of Cocoa APIs now already returns the correct type.

EDIT: (october 18, 2016)
Note that, as of Swift 3.0, Objective-C ids are now imported as Any, not anymore as AnyObject.