The Universe must contain some form of dark matter (DM), based on many astrophysical and cosmological observations. The Standard Model of particle physics (SM) does not contain a DM candidate. We may extend the SM by the addition of a single large electroweak complex scalar multiplet. These multiplets may have at most 8 members. Based on the isospin and hypercharge assignments, and the form of the scalar potential, the models we examine in this thesis have either 6 or 8 members. One of the newly-introduced particles is a DM candidate. We use the following theoretical considerations and experimental results to constrain the parameter space of these models: perturbative unitarity of scattering; stability of the scalar potential; direct searches for new physics at the LHC; electroweak observables (STU); and decays of the Higgs boson. We then compute the relic abundance in the models to determine the viability of our DM candidate. Finally, we examine the prospects of discovery at direct detection experiments. We find that these models may form part of the total DM content in the mass range 80-1000 GeV, and will begin to be probed in the direct detection experiments currently under construction. For the mass range > 5 TeV, we must also consider the effects of co-annihilation and Sommerfeld enhancement. In this range, our DM candidate may be all of the DM, but will require a subsequent generation of direct detection experiments to probe its parameter space.