NSRange to Range

As of Swift 4 (Xcode 9), the Swift standard
library provides methods to convert between Swift string ranges
(Range<String.Index>) and NSString ranges (NSRange).
Example:

let str = "a👿b🇩🇪c"
let r1 = str.range(of: "🇩🇪")!

// String range to NSRange:
let n1 = NSRange(r1, in: str)
print((str as NSString).substring(with: n1)) // 🇩🇪

// NSRange back to String range:
let r2 = Range(n1, in: str)!
print(str[r2]) // 🇩🇪

Therefore the text replacement in the text field delegate method
can now be done as

func textField(_ textField: UITextField,
               shouldChangeCharactersIn range: NSRange,
               replacementString string: String) -> Bool {

    if let oldString = textField.text {
        let newString = oldString.replacingCharacters(in: Range(range, in: oldString)!,
                                                      with: string)
        // ...
    }
    // ...
}

---

(Older answers for Swift 3 and earlier:)

As of Swift 1.2, String.Index has an initializer

init?(_ utf16Index: UTF16Index, within characters: String)

which can be used to convert NSRange to Range<String.Index> correctly
(including all cases of Emojis, Regional Indicators or other extended
grapheme clusters) without intermediate conversion to an NSString:

extension String {
    func rangeFromNSRange(nsRange : NSRange) -> Range<String.Index>? {
        let from16 = advance(utf16.startIndex, nsRange.location, utf16.endIndex)
        let to16 = advance(from16, nsRange.length, utf16.endIndex)
        if let from = String.Index(from16, within: self),
            let to = String.Index(to16, within: self) {
                return from ..< to
        }
        return nil
    }
}

This method returns an optional string range because not all NSRanges
are valid for a given Swift string.

The UITextFieldDelegate delegate method can then be written as

func textField(textField: UITextField, shouldChangeCharactersInRange range: NSRange, replacementString string: String) -> Bool {

    if let swRange = textField.text.rangeFromNSRange(range) {
        let newString = textField.text.stringByReplacingCharactersInRange(swRange, withString: string)
        // ...
    }
    return true
}

The inverse conversion is

extension String {
    func NSRangeFromRange(range : Range<String.Index>) -> NSRange {
        let utf16view = self.utf16
        let from = String.UTF16View.Index(range.startIndex, within: utf16view) 
        let to = String.UTF16View.Index(range.endIndex, within: utf16view)
        return NSMakeRange(from - utf16view.startIndex, to - from)
    }
}

A simple test:

let str = "a👿b🇩🇪c"
let r1 = str.rangeOfString("🇩🇪")!

// String range to NSRange:
let n1 = str.NSRangeFromRange(r1)
println((str as NSString).substringWithRange(n1)) // 🇩🇪

// NSRange back to String range:
let r2 = str.rangeFromNSRange(n1)!
println(str.substringWithRange(r2)) // 🇩🇪

Update for Swift 2:

The Swift 2 version of rangeFromNSRange() was already given
by Serhii Yakovenko in this answer, I am including it
here for completeness:

extension String {
    func rangeFromNSRange(nsRange : NSRange) -> Range<String.Index>? {
        let from16 = utf16.startIndex.advancedBy(nsRange.location, limit: utf16.endIndex)
        let to16 = from16.advancedBy(nsRange.length, limit: utf16.endIndex)
        if let from = String.Index(from16, within: self),
            let to = String.Index(to16, within: self) {
                return from ..< to
        }
        return nil
    }
}

The Swift 2 version of NSRangeFromRange() is

extension String {
    func NSRangeFromRange(range : Range<String.Index>) -> NSRange {
        let utf16view = self.utf16
        let from = String.UTF16View.Index(range.startIndex, within: utf16view)
        let to = String.UTF16View.Index(range.endIndex, within: utf16view)
        return NSMakeRange(utf16view.startIndex.distanceTo(from), from.distanceTo(to))
    }
}

Update for Swift 3 (Xcode 8):

extension String {
    func nsRange(from range: Range<String.Index>) -> NSRange {
        let from = range.lowerBound.samePosition(in: utf16)
        let to = range.upperBound.samePosition(in: utf16)
        return NSRange(location: utf16.distance(from: utf16.startIndex, to: from),
                       length: utf16.distance(from: from, to: to))
    }
}

extension String {
    func range(from nsRange: NSRange) -> Range<String.Index>? {
        guard
            let from16 = utf16.index(utf16.startIndex, offsetBy: nsRange.location, limitedBy: utf16.endIndex),
            let to16 = utf16.index(utf16.startIndex, offsetBy: nsRange.location + nsRange.length, limitedBy: utf16.endIndex),
            let from = from16.samePosition(in: self),
            let to = to16.samePosition(in: self)
            else { return nil }
        return from ..< to
    }
}

Example:

let str = "a👿b🇩🇪c"
let r1 = str.range(of: "🇩🇪")!

// String range to NSRange:
let n1 = str.nsRange(from: r1)
print((str as NSString).substring(with: n1)) // 🇩🇪

// NSRange back to String range:
let r2 = str.range(from: n1)!
print(str.substring(with: r2)) // 🇩🇪