Declare a delegate type in Typescript

You can create something like a delegate by using a type alias:

type MyDelegate = (input: string) => void;

which defines a type name for a function pointer, just as delegates do in C#. The following example uses it in combination with generic type parameters:

type Predicate<T> = (item: T) => boolean;

export class List<T> extends Array<T> {
    constructor(...items: T[]){
        super();
        for(let i of items || []){
            this.push(i);
        }
    }
    public hasAny(predicate?: Predicate<T>): boolean {
        predicate = predicate || (i => true)
        for(let item of this) {
            if(predicate(item)) return true;
        }
        return false;
    }
}

In TypeScript, interfaces can have call signatures. In your example, you could declare it like this:

interface Greeter {
    (message: string): void;
}

function sayHi(greeter: Greeter) {
    greeter('Hello!');
}

sayHi((msg) => console.log(msg)); // msg is inferred as string

Type definition for a callable expression (this is a draft ok, for humans... not a BNF or anything formal):

callableType: (paramsDef) => returnType

paramsDef:    MULTIPLE paramDef SEPARATED BY ,

paramDef:     EITHER   paramName: paramType
                  OR   optionalParamName?: paramTypeWhenDefined
                  OR   ...manyParamName: eachParamType[]

Example:

var func = something as ((...x: any[]) => any);

Then you can:

var result = func("a", "b", 2);

Five years and many, many TS versions later I find myself using a simpler type definition for declaring function types:

type Greeter = (msg: string) => void;
const someGreeter: Greeter = (msg: string) => `Hi there with ${msg}`;