Higgs Physics at Future Colliders: recent theoretical developments

I review the physics of the Higgs sector in the Standard Model and its minimal supersymmetric extension, the MSSM. I will discuss the prospects for discovering the Higgs particles at the upgraded Tevatron, at the Large Hadron Collider, and at a future high--energy $e^+e^-$ linear collider with centre--of--mass energy in the 350--800 GeV range, as well as the possibilities for studying their fundamental properties. Some emphasis will be put on the theoretical developments which occurred in the last two years.