How can I cast an NSMutableArray to a Swift array of a specific type?

You can make this work with a double downcast, first to NSArray, then to [String]:

var strings = myObjcObject.getStrings() as NSArray as [String]

Tested in a Playground with:

import Foundation

var objCMutableArray = NSMutableArray(array: ["a", "b", "c"])
var swiftArray = objCMutableArray as NSArray as [String]

Update:

In later versions of Swift (at least 1.2), the compiler will complain about as [String]. Instead you should use an if let with a conditional downcast as?:

import Foundation

var objCMutableArray = NSMutableArray(array: ["a", "b", "c"])
if let swiftArray = objCMutableArray as NSArray as? [String] {
    // Use swiftArray here
}

If you are absolutely sure that your NSMutableArray can be cast to [String], then you can use as! instead (but you probably shouldn't use this in most cases):

import Foundation

var objCMutableArray = NSMutableArray(array: ["a", "b", "c"])
var swiftArray = objCMutableArray as NSArray as! [String]

compactMap is your friend in Swift 4.1 and above, as well as in Swift 3.3-3.4 for that matter. This means that you don't have any double or forced casting.

let mutableArray = NSMutableArray(array: ["a", "b", "c"])
let swiftArray: [String] = mutableArray.compactMap { $0 as? String }

In previous versions of Swift, 2.0-3.2 and 4.0, you'll want to use flatMap for this purpose. The usage is the same as compactMap:

let swiftArray: [String] = mutableArray.flatMap { $0 as? String }

With Swift 1.2 the following will work:

let mutableArray = NSMutableArray(array: ["a", "b", "c"])
let swiftArray = NSArray(array: mutableArray) as? [String]