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Hydrodynamics is a universal theory of thermalization in chaotic many-body systems, both classical and quantum. It is minimally described by symmetries and conservation laws under the assumption of local thermodynamic equilibrium. Despite its universal character, there are many important scenarios in which the usual hydrodynamic picture can break down, giving rise to qualitatively distinct dynamical regimes with anomalous scaling exponents. In this talk, I will discuss two such scenarios. In the first case, I describe thermalization of two-dimensional spin systems and argue that such systems exhibit slow thermalization due to the existence of slow modes with quasi-long-range character. Such modes give rise to a universal self-similar regime with anomalous scaling exponents. In the second case, I discuss microscopic models with kinematic constraints---fractonic fluids---in which the hydrodynamic description is intrinsically unstable, leading to a higher-dimensional analogue of the Kardar-Parisi-Zhang (KPZ) universality class. Our understanding of anomalous scaling regimes beyond hydrodynamics can help us create more systematic ways to classify and characterize universality in systems out of equilibrium.