What does the error "JSX element type '...' does not have any construct or call signatures" mean?

This is a confusion between constructors and instances.

Remember that when you write a component in React:

class Greeter extends React.Component<any, any> {
    render() {
        return <div>Hello, {this.props.whoToGreet}</div>;
    }
}

You use it this way:

return <Greeter whoToGreet='world' />;

You don't use it this way:

let Greet = new Greeter();
return <Greet whoToGreet='world' />;

In the first example, we're passing around Greeter, the constructor function for our component. That's the correct usage. In the second example, we're passing around an instance of Greeter. That's incorrect, and will fail at runtime with an error like "Object is not a function".

---

The problem with this code

function renderGreeting(Elem: React.Component<any, any>) {
    return <span>Hello, <Elem />!</span>;
}

is that it's expecting an instance of React.Component. What you want is a function that takes a constructor for React.Component:

function renderGreeting(Elem: new() => React.Component<any, any>) {
    return <span>Hello, <Elem />!</span>;
}

or similarly:

function renderGreeting(Elem: typeof React.Component) {
    return <span>Hello, <Elem />!</span>;
}

If you want to take a component class as a parameter (vs an instance), use React.ComponentClass:

function renderGreeting(Elem: React.ComponentClass<any>) {
    return <span>Hello, <Elem />!</span>;
}

When I'm converting from JSX to TSX and we keep some libraries as js/jsx and convert others to ts/tsx I almost always forget to change the js/jsx import statements in the TSX\TS files from

import * as ComponentName from "ComponentName";

to

import ComponentName from "ComponentName";

If calling an old JSX (React.createClass) style component from TSX, then use

var ComponentName = require("ComponentName")