Riemann surfaces are the simplest spaces to which one can extend the techniques of complex geometry and analysis. They were introduced by Bernhard Riemann in the nineteenth century, and have since become one of the most prolific and central topics in Mathematics, whose ideas and problems keep resurfacing where one least expects them to.In this talk we will try to see how this happens, and why Riemann surfaces tell us stories of mathematical unity, by bringing together, sometimes in the most uncanny ways, algebra, geometry and analysis. Time permitting, we hope to provide a glimpse at the relations between Riemann Surfaces and hyperbolic and algebraic geometry, field, number and group theory and dynamical systems.