Hydrogels attract great attention as biomaterials as a result of their soft and wet nature, similar to that of biological tissues. Recent inventions of several tough hydrogels show their potential as structural biomaterials, such as cartilage. Any given application, however, requires a combination of mechanical properties including stiffness, strength, toughness, damping, fatigue resistance and self-healing, along with biocompatibility. This combination is rarely realized. Here, we report that polyampholytes, polymers bearing randomly dispersed cationic and anionic repeat groups, form tough and viscoelastic hydrogels with multiple mechanical properties. The randomness makes ionic bonds of a wide distribution of strength. The strong bonds serve as permanent crosslinks, imparting elasticity, whereas the weak bonds reversibly break and re-form, dissipating energy. These physical hydrogels of supramolecular structure can be tuned to change multiple mechanical properties over wide ranges by using diverse ionic combinations. This polyampholyte approach is synthetically simple and dramatically increases the choice of tough hydrogels for applications.