I don't understand when to use AnyObject and when to user Any in Swift.
My case, I've a Dictionary
AnyObject is only for reference types (classes),
Any is for both value and reference types.
So you should go for
Type Casting for Any and AnyObject
Swift provides two special type aliases for working with non-specific types:
AnyObject can represent an instance of any class type. Any can represent an instance of any type at all, apart from function types.
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/prerelease/mac/documentation/Swift/Conceptual/Swift_Programming_Language/TypeCasting.html#//apple_ref/doc/uid/TP40014097-CH22-XID_503
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.
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.