We present the results of the first systematic X-ray spectroscopy of 11 supernova remnants (SNRs) in the Magellanic Clouds. We constructed self-consistent non-equilibrium ionization SNR models utilizing self-similar solutions for the dynamical evolution of SNRs, and then applied the resulting spectral models to the data obtained by the ASCA satellite. For middle-aged evolved remnants in the LMC (3000-10000 years old; N23, N49, N63A, DEM71, N132D, 0453-68.5, and N49B), our Sedov model well describe their ASCA spectra, allowing us to derive accurate values for their physical quantities, i.e., ages, densities, initial explosion energies, and metal abundances. We find no evidence for enrichment by supernova ejecta in any of these remnants; rather each appears to be dominated by swept-up interstellar material. Using the SNR sample we determine that the mean LMC gas-phase abundances of the astrophysically common elements from oxygen to iron are lower than the cosmic values by factors of 2-4. Overall these results are consistent with previous ones based on optical data, but do not show the anomalous overabundance of magnesium and silicon seen by Russell & Dopita(1992). Our independent measurement of the current chemical composition of the LMC provides a significant new constraint on the chemical evolution and starformation history of the Cloud. For young remnants (<~ 2000 years old), we find their ejecta-dominated X-ray spectra, and then quantitatively confirm the significant overabundance of several elements. The three Type Ia remnants in the LMC (N103B, 0509-67.5, and 0519-69.0) show the overabundance of silicon, sulfur, argon, calcium, and iron. On the other hand, a Type II remnant in the SMC (0102-72.3) shows the overabundance of oxygen, neon, and magnesium. This clear-cut spectral difference resulted in X-ray typing of supernovae from their remnants (Hughes et al. 1995). We also find that the phenomenological models frequently used in the previous X-ray spectral analysis for many SNRs are likely to lead us to an erroneous abundance estimation (up to factor 10). On the other hand, we argue that the abundances we have measured here do not strongly depend on the details of an assumed hydrodynamical structure of SNRs, as far as one assumes homogeneous elemental distributions and takes account of a reasonable internal structure of SNRs unlike the phenomenological models. The energy non-equipartition between electrons and ions is strongly indicated for 0519-69.0.