The discovery of the Higgs boson marks the first direct probe into the
mechanism of electroweak symmetry breaking. All evidence currently
points towards the fact that electroweak symmetry is broken by at
least one fundamental scalar, and naturalness remains the most
compelling reason to expect additional degrees of freedom at the weak
scale. This talk will describe some ideas for how to utilize powerful
and proven experimental techniques, in conjunction with the fact that
observables directly related to the Higgs boson are now experimentally
accessible, to make concrete statements about the existence of any
such new degrees of freedom.