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The dark photon is one of the simplest and most versatile extensions of the Standard Model, with markedly different phenomenology in the massless and massive limits. In this talk, I will discuss two recent developments in dark photon physics. First, I will consider light minicharged particles associated with a massless dark photon. In this framework, anomaly cancellation imposes highly nontrivial constraints on chiral charge assignments, and these conditions can be related to the Prouhet–Tarry–Escott problem in number theory. I will show that this framework predicts a doublet structure of the minicharged particles. I will then turn to the massive case and discuss light dark photon dark matter, focusing on a general cosmological bound in scenarios where the dark photon mass is generated through spontaneous symmetry breaking. Together, these examples illustrate how theoretical consistency, mathematical structure, and cosmological constraints shape the viable parameter space of dark photon models.