Variational quantum algorithms (VQAs) constitute a class of hybrid quantum-classical algorithms that are envisioned as appropriate for noisy intermediate scale quantum processors. The majority of VQAs focus on quantum simulation, and particularly finding properties of many-body quantum systems, such as the ground state energies of complicated molecules. Other problems, such as optimization and machine learning are also being explored with this approach. In VQAs, the quantum processor is where the quantum state is variationally prepared and where measurements are made, while the classical computer performs optimization. I will present our work on ADAPT-VQE and its variations for designing efficient, compact ansatze both for many-body and for optimization problems.