Measurements of the deuteron elastic magnetic structure function B(Q') are reported at squared four-momentum transfer values 1.20~ Qi ~ 2.77 (GeV/c)'. Also reported are values for the proton magnetic form factor Gstr(Q2) at 11 Q values between 0.49 and 1.75 (GeV/c) . The data were obtained using an electron beam of 0.5 to 1.3 GeV. Electrons backscattered near 180' were detected in coincidence with deuterons or protons recoiling near 0 in a large solid-angle double-arm spectrometer system. The data for B(Q ) are found to decrease rapidly from Q =1.2 to 2 (GeV/c), and then rise to a secondary maximum around Q'=2. 5 (GeV/c)'. Reasonable agreement is found with several different models, including those in the relativistic impulse approximation, nonrelativistic calculations that include meson-exchange currents, isobar configurations, and six-quark configurations, and one calculation based on the Skyrme model. All calculations are very sensitive to the choice of deuteron wave function and nucleon form factor parametrization. The data for GMr(Q') are in good agreement with the empirical dipole fit.