We apply theoretical and experimental constraints to models that extend the Standard Model (SM) with scalar electroweak SU(2) multiplets of isospin T>1/2. Using tree-level perturbative unitarity, we determine that such multiplets may have a maximum weak isospin of 7/2 (4) if they are complex (real). We also determine an upper bound on the hypercharge of a scalar SU(2) multiplet as a function of its size n=2T+1. Using these bounds, we study two classes of models which extend the SM by a single scalar SU(2) multiplet of isospin greater or equal to 2 and which preserve an additional U(1) or Z2 symmetry. By applying bounds from precision electroweak measurements, Higgs decays to two photons, and dark matter direct-detection, we determine that the U(1) models with isospin T=5/2,3,7/2 are excluded. The Z2 models remain viable, but can account for at most 1% of the dark matter in the universe at masses below 1 TeV.
We also study the most general scalar potential of the Georgi-Machacek model, which extends the SM with two electroweak scalar triplets. We show that this model possesses a decoupling limit, and that in this limit the deviations of the Higgs couplings to vector bosons from their SM values may be larger and of opposite sign to those of the Two Higgs Doublet Model. We apply constraints from perturbativity, vacuum stability, electroweak precision observables, and B-physics, and demonstrate their effects on the model parameter space with numerical scans. We show that --- subject to these constraints --- the Georgi-Machacek (GM) model can simultaneously enhance all of the Higgs couplings to Standard Model particles by up to 18%, corresponding to a 39% enhancement of all Higgs production modes. Finally, we derive general formulae for loop decays of the GM scalars to two photons, Z+photon, and W+photon, and consider their effects on the scalar branching ratios in a benchmark case.