Chern insulators, which are the lattice analogs of the quantum Hall states, can potentially manifest high-temperature topological orders at zero magnetic field to enable next-generation topological quantum devices 1-4. To date, integer Chern insulators have been experimentally demonstrated in several systems at zero magnetic field 3, 5-11, but fractional Chern insulators have been reported only in graphene-based systems under a finite magnetic field 12, 13. The emergence of semiconductor moiré materials 14, 15, which support tunable topological flat bands 16, 17, opens a new opportunity to realize fractional Chern insulators 18-20. Here, we report the observation of both integer and fractional Chern insulators at zero magnetic field in small-angle twisted bilayer MoTe2 by combining the local electronic compressibility and magneto-optical measurements. At hole filling factor v= 1 and 2/3, the system is incompressible and spontaneously breaks time reversal symmetry. We determine the Chern number to be 1 and 2/3 for the v=1 and v=2/3 gaps, respectively, from their dispersion in filling factor with applied magnetic field using the Streda formula. We further demonstrate electric-field-tuned topological phase transitions involving the Chern insulators. Our findings pave the way for demonstration of quantized fractional Hall conductance and anyonic excitation and braiding 21 in semiconductor moiré materials.