There has been a good deal of work on algorithms to simulate quantum many-body systems with fault-tolerant quantum computers- those with full error correction. Fault-tolerant quantum computers of scale requisite to achieve computational advantage for these problems are likely over a decade away. Moreover, devices that we can build in the near term, called Noisy Intermediate Scale Quantum computers (NISQ), have too much noise to implement the long circuits required by these algorithms. We review heuristic, short-depth quantum algorithms more suited to NISQ computers; specifically, their scaling properties when applied to electronic and nuclear structure calculations, including Hamiltonian complexity with particle number, ansatz state preparation, convergence, and noise. We will present examples of actual quantum structure calculations with NISQ computers, as well as a newly-developed error mitigation technique that significantly improves accuracy. We end with an outlook for "advantage" - when NISQ systems might excel conventional HPC approaches for comparable problems.