This review discusses the application of machine learning techniques, specifically neural networks, to the numerical study of quantum many-body systems. It highlights the potential of neural quantum states (NQS) in accurately modeling complex many-body phenomena in spin, fermionic, and qubit systems. The review covers the central equations of variational Monte Carlo (VMC) approaches, including ground state search, time evolution, and overlap optimization, and discusses data-driven tasks such as quantum state tomography. Key topics include the geometry of the variational manifold, practical implementation challenges, and recent results in first-principles ground-state and real-time calculations. The review also provides a pedagogical introduction to the main concepts and theory behind using neural network approximations for many-body wave functions, their optimization, and applications.This review discusses the application of machine learning techniques, specifically neural networks, to the numerical study of quantum many-body systems. It highlights the potential of neural quantum states (NQS) in accurately modeling complex many-body phenomena in spin, fermionic, and qubit systems. The review covers the central equations of variational Monte Carlo (VMC) approaches, including ground state search, time evolution, and overlap optimization, and discusses data-driven tasks such as quantum state tomography. Key topics include the geometry of the variational manifold, practical implementation challenges, and recent results in first-principles ground-state and real-time calculations. The review also provides a pedagogical introduction to the main concepts and theory behind using neural network approximations for many-body wave functions, their optimization, and applications.