We numerically study the dynamics of a driven magnetic domain wall through a disordered medium, which is modeled as a random field disorder. The domain wall is driven by an external force and is fluctuated by the disorder and the thermal noise. In particular, we focus on some scaling exponents characterizing the depinning behavior at zero temperature. Performing large scale simulations, we obtain a velocity exponent theta; = 0.76 ± 0.02, a dynamical exponent beta; = 1.56 ± 0.02, and a roughness exponent ζ = 0.65 ± 0.03. The thermal depinning phenomenon is also discussed.