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Typically discussed in the context of optics, caustics are envelopes of classical trajectories (rays) where the density of states diverges, resulting in pronounced observable features such as bright points, curves, and extended networks of patterns. Here, we generate caustics in the matter waves of an atom laser, providing a striking experimental example of catastrophe theory applied to atom optics in an accelerated (gravitational) reference frame. We showcase caustics formed by individual attractive and repulsive potentials, and present an example of a network generated by multiple potentials. Exploiting internal atomic states, we demonstrate fluid-flow tracing as another tool of this flexible experimental platform. The effective gravity experienced by the atoms can be tuned with magnetic gradients, forming caustics analogous to those produced by gravitational lensing. From a more applied point of view, atom optics affords perspectives for metrology, atom interferometry, and nanofabrication. Caustics in this context may lead to quantum innovations as they are an inherently robust way of manipulating matter waves.