Mono- and bilayer graphene have generated tremendous excitement owing to their unique and potentially useful electronic properties. Suspending single-layer graphene flakes above the substrate has been shown to greatly improve sample quality, yielding high-mobility devices with little charge inhomogeneity. Here we report the fabrication of suspended bilayer graphene devices with very little disorder. We observe quantum Hall states that are fully quantized at a magnetic field of 0.2T, as well as broken-symmetry states at intermediate filling factors ν=0, +/-1, +/-2 and +/-3. In the ν=0 state, the devices show extremely high magnetoresistance that scales as magnetic field divided by temperature. This resistance is predominantly affected by the perpendicular component of the applied field, and the extracted energy gap is significantly larger than expected for Zeeman splitting. These findings indicate that the broken-symmetry states arise from many-body interactions and underscore the important part that Coulomb interactions play in bilayer graphene.