The Higgs boson plays a central role in the Standard Model of Elementary Particle Physics, through spontaneous electroweak symmetry breaking and by providing masses to fermions. Similarly, the Higgs boson plays a special role in many extensions of the Standard Model. In particular, the Higgs boson represents an extraordinary window to search for extensions of the Standard Model that feature a Dark Matter particle candidate. I will review such searches for Dark Matter through the Higgs window at colliders, with a particular focus on the contribution of our group using the ATLAS detector at the Large Hadron Collider at CERN, and will give an outlook into the future at the High-Luminosity LHC.