In high-quality c-axis-oriented MgB2 thin films, we observed high critical current densities ( J(c)) of approximately 16 MA/cm(2) at 15 K under self-fields comparable to those of cuprate high-temperature superconductors. The extrapolated value of J(c) at 5 K was estimated to be approximately 40 MA/cm(2). For a magnetic field of 5 T, a J(c) of approximately 0.1 MA/cm(2) was detected at 15 K, suggesting that this compound would be a very promising candidate for practical applications at high temperature and lower power consumption. The vortex-glass phase is considered to be a possible explanation for the observed high current-carrying capability.