If fundamental scalar bosons exist, a natural implication of theories based on supersymmetry or superstrings, Higgs boson masses are likely to be of order m Z within roughly a factor of two. We term this the intermediate mass range. We study how such Higgs bosons could be detected at e +e − and, particularly, at hadron colliders. At a hadron collider it appears to be necessary to use rare decay modes of the Higgs boson, and we examine H 0→ γγ, τ + τ −, Z γ , Zℓ +ℓ −, Θγ,and b b . Since charged scalars also occur in all theories beyond the minimal Standard Model, we examine ways to find the H ±, concentrating on H ±→ τ ± ν, W ±H 0, W ±H 0 γ, W ± γ, and W ±Z. Because obtaining data about the spectrum of Higgs bosons is so essential, our approach is to ask what facilities would be sufficient to cover the entire intermediate mass range.