Quantum computer could solve classically intractable problems. Since realizing a universal quantum computer is challenging with current technology, a more practical question before having a fully-fledged quantum computer is what we can do with current and near-term quantum hardware. Focusing on the noisy-intermediate-scaled-quantum regime, we introduce variational quantum algorithms for solving static and dynamic problems of many-body physics. We show how to suppress device errors due to implementation imperfection on both digital and analog quantum computers. The algorithms are also applicable to other tasks, including quantum machine learning, quantum sensing, and quantum error correction. With the rapid development of quantum hardware, error-mitigated variational quantum algorithms may finally enable genuine quantum advantage demonstration in the noisy-intermediate-scaled quantum era.