Hydrodynamic excitations corresponding to sound and diffusive modes in fluids are characterised by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by infinite power series in spatial momenta. I will discuss how the introduction of a new concept of the hydrodynamic complex spectral curve in the space of complexified frequency and spatial momentum—the concept otherwise known from algebraic geometry---can be used to prove general properties about hydrodynamics, including its finite radius of convergence. When the infinite series are resummed, they exhibit a fascinating, recently-discovered phenomenon of pole-skipping, which enables us to analyse the underlying, microscopic quantum many-body chaos in the system. Throughout my talk, I will use gauge-gravity duality as a tool to explicitly show these phenomena in holographic systems and discuss what their implications are for the dual gravity theory.