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On occasion, phenomena encountered in other fields of physics find themselves an equivalence in accelerator physics. Caustics are one such example. Within the fields of X-ray optics and electron microscopy, caustics are commonly recognized and are well-understood. The same effect can be seen in various accelerator physics scenarios. Caustics are a form of ‘natural focusing’, whereby particle trajectories coalesce to form regions of greatly enhanced charge density. By their nature, caustics are discontinuities that result from small continuous perturbations of an input, and under certain conditions, small density modulations will reliably produce striking changes in the corresponding output current profile. This talk will detail some examples and methods that use knowledge of caustics to shape the longitudinal beam shape to our advantage. Such applications include: i) suppression of head and tail current spikes encountered in strong bunch compression, to minimise coherent synchrotron radiation (CSR)-induced emittance growth in FELs, ii) longitudinal phase space management as a key feature of recirculating machines, and iii) as a method to investigate microbunching instability. I will also share experimental measurements of the caustic nature of charged particle trajectories that were conducted at the MAX IV (Lund, Sweden), which showed how the natural amplification of small perturbations can make caustics an attractive diagnostic tool, and effective tool for characterising bunch compressors.