Magnetic confinement nuclear fusion holds great promise as a source of clean and sustainable energy for the future. However, achieving net energy from fusion reactors requires a more profound understanding of the underlying physics and the development of efficient control strategies. Plasma diagnostics are vital to these efforts, but accessing local information often involves solving very ill-posed inverse problems. Regrettably, many of the current approaches for solving these problems rely on simplifying assumptions, sometimes inaccurate or not completely verified, with consequent imprecise outcomes. In order to overcome these challenges, the present study suggests employing physics-informed neural networks (PINNs) to tackle inverse problems in tokamaks. PINNs represent a type of neural network that is versatile and can offer several benefits over traditional methods, such as their capability of handling incomplete physics equations, of coping with noisy data, and of operating mesh-independently. In this work, PINNs are applied to three typical inverse problems in tokamak physics: equilibrium reconstruction, interferometer inversion, and bolometer tomography. The reconstructions are compared with measurements from other diagnostics and correlated phenomena, and the results clearly show that PINNs can be easily applied to these types of problems, delivering accurate results. Furthermore, we discuss the potential of PINNs as a powerful tool for integrated data analysis. Overall, this study demonstrates the great potential of PINNs for solving inverse problems in magnetic confinement thermonuclear fusion and highlights the benefits of using advanced machine learning techniques for the interpretation of various plasma diagnostics.