Neural-network quantum states (NQS) have become a powerful tool for simulating many-body quantum systems. This review discusses the application of machine learning techniques to improve the accuracy and efficiency of numerical many-body methods. NQS are parametric wave functions represented by neural networks, which can efficiently approximate complex quantum states. The review covers the central equations of variational Monte Carlo (VMC) methods, including ground state search, time evolution, and overlap optimization, as well as data-driven tasks like quantum state tomography. It emphasizes the geometry of the variational manifold and challenges in practical implementations. Recent results in first-principles ground-state and real-time calculations are also discussed. The review introduces the general framework of VMC, the need for Monte Carlo techniques to estimate expectation values and gradients, and four scenarios for variational parameter optimization: ground state search, time evolution, overlap optimization, and maximum likelihood estimation. These techniques form the basis for applications in many-body systems. The review also discusses the application of NQS to quantum state tomography and the imposition of symmetries in NQS calculations. The review concludes with an overview of applications in spin systems, fermionic systems, and quantum state tomography. The review highlights the potential of NQS in describing challenging many-body problems, including interacting spins and fermions, and their applications in time evolution, quantum circuit simulation, and quantum state tomography. The review also discusses the challenges in optimizing NQS, including the computational overhead for large parameter sets and the need for efficient gradient-based optimization methods. The review concludes with an overview of the applications of NQS in various many-body systems and the importance of symmetry considerations in NQS calculations.Neural-network quantum states (NQS) have become a powerful tool for simulating many-body quantum systems. This review discusses the application of machine learning techniques to improve the accuracy and efficiency of numerical many-body methods. NQS are parametric wave functions represented by neural networks, which can efficiently approximate complex quantum states. The review covers the central equations of variational Monte Carlo (VMC) methods, including ground state search, time evolution, and overlap optimization, as well as data-driven tasks like quantum state tomography. It emphasizes the geometry of the variational manifold and challenges in practical implementations. Recent results in first-principles ground-state and real-time calculations are also discussed. The review introduces the general framework of VMC, the need for Monte Carlo techniques to estimate expectation values and gradients, and four scenarios for variational parameter optimization: ground state search, time evolution, overlap optimization, and maximum likelihood estimation. These techniques form the basis for applications in many-body systems. The review also discusses the application of NQS to quantum state tomography and the imposition of symmetries in NQS calculations. The review concludes with an overview of applications in spin systems, fermionic systems, and quantum state tomography. The review highlights the potential of NQS in describing challenging many-body problems, including interacting spins and fermions, and their applications in time evolution, quantum circuit simulation, and quantum state tomography. The review also discusses the challenges in optimizing NQS, including the computational overhead for large parameter sets and the need for efficient gradient-based optimization methods. The review concludes with an overview of the applications of NQS in various many-body systems and the importance of symmetry considerations in NQS calculations.