This thesis examines the application of physics-informed neural networks to solve free-surface flow problems modeled with the shallow water equations. Physics-informed neural networks allow training of a surrogate model that resembles the latent solution of an underlying partial differential equation, without using any training data sampled from experiments or numerical simulations. The shallow water equations are an approximation of the Navier stokes equations and serve as a model to many environmental flow problems including dam-breaks, floods, and tsunami propagation. The equations form a non-linear system of hyperbolic partial differential equations that describe the evolution of a fluid's depth and momentum through time. Contrary to other models for free-surface flow, where the exact location of the free surface is only given implicitly as an isosurface and needs reconstruction, here, the depth directly yields its location. One characteristic of the shallow water equations is the formation of steep wavefronts and discontinuities. The thesis examines four state-of-the-art techniques to improve accuracy and training speed and discusses their behavior on three initial value problems. These include the famous idealized dam-break and two depth perturbations, one above a flat and one above varying bathymetry. For each of the scenarios, an inspection of suitable network architectures was considered. Additionally, three different formulations of the physics-informed neural network are presented and tested, where one approach implicitly fulfills the mass conservation and thus eliminates one equation of the system. The results show, that it is possible to train a surrogate model with a relative L^2 error of less than 10^(-4) compared to a solution computed by a high-resolution numerical solver in case of a moderate steepening of wavefronts. A relative error close to 10^(-3) can be achieved for the dam break problem, where the initial conditions are discontinuous, and the solution contains shocks that propagate over time. Additionally, it shows that training with bathymetry is possible and the learned depth approximates the varying underground without any noticeable difference.