I know that it used to be said, in praise by some and as criticism by others, that Number Theory had no applications. Now it is used in cryptography and Quantum Theory.
Since the mathematics that explain physical systems must be invented before those systems can be so described, the invention/discovery of maths must always precede its utility and therefore the newest math will always be without an application.
However, are there still any major genres of mathematics that lack any applications? If so, what are they?