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?
Whereas most mathematics did have direct applications in the past (geometry and calculus), now there are branches that have applications on other branches of mathematics. I think about logic, model theory, set theory or computability. Most of this branches have applications and consequences on other mathematics, but not really or directly on any "physical problems".
For example, Continuum hypothesis or the fact that there are non comparable Turing degrees have no direct consequences on physical problem but they speak (loud) about what are the mathematics themselves.
I think this is the revolution of the previous century : Mathematics became a science.