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Fluctuations are ubiquitous in non-equilibrium systems. While their mean behavior in hydrodynamic limits is described by partial differential equations, the fluctuations around this mean can be described in terms of large deviations estimates. The analysis of such large deviations has led to the so-called macroscopic fluctuation theory, a general framework for non-equilibrium statistical mechanics. In this talk, we will survey this link between hydrodynamic limits, large deviations, and PDEs for the example of the zero range process. We will demonstrate how the task of proving large deviations estimates leads to the analysis of PDEs with irregular coefficients in critical spaces and to Gamma convergence. Finally, by developing a nonlinear extension of the DiPerna-Lions theory of renormalized solutions, we solve a long-standing open problem in the proof of a full large deviation principle for the zero-range process.