We observed the conduction electron spin resonance (CESR) in fine powders of MgB2 both in the superconducting and normal states. The Pauli susceptibility is chi(s) = 2.0 x 10(-5) emu/mole in the temperature range of 450 to 600 K. The spin relaxation rate has an anomalous temperature dependence. The CESR measured below T(c) at several frequencies suggests that MgB2 is a strongly anisotropic superconductor with the upper critical field, H(c2), ranging between 2 and 16 T. The high-field reversible magnetization data of a randomly oriented powder sample are well described assuming that MgB2 is an anisotropic superconductor with H(ab)(c2)/H(c)(c2) approximately 6-9.