Improvements in the control and coherence of artificial atoms built from superconducting circuits have enabled the development of noisy processors with over 50 qubits, and the exploration of problems addressable by these devices that are intractable to classical computation. In this talk, I shall present a brief summary of superconducting qubit technology and highlight some of the dominant errors and outstanding challenges. I shall then discuss results from small-scale demonstrations of algorithms for quantum simulation. These experiments highlight the detrimental effect of decoherence and measurement errors on computations with noisy quantum processors. While this can be remedied, in theory, with quantum error correction, the resources required for its physical realization are prohibitively large for the near term. In this context, I shall introduce “error mitigation” techniques that enable access to noise-free estimates of expectation values for short depth quantum circuits, without requiring any additional hardware resources.