Yilin Wang of ETH Zurich delivers this Bourgain Lecture at the Institute for Advanced Study on the mathematics of random curves. She starts with universality in probability theory, the way random walks converge to Brownian motion regardless of their underlying distribution, and traces how this phenomenon gives rise to the Schramm-Loewner Evolution (SLE) as the universal scaling limit of interfaces in two-dimensional conformally invariant systems. She then turns to a separate structure, the Weil-Petersson Teichmuller space of regular simple curves, which carries an essentially unique Kahler geometry. The lecture's core is the Loewner energy, the bridge connecting these two seemingly unrelated worlds through group and Kahler structures defined via infinitesimal curve variations. Wang closes with recent work linking Loewner energy to renormalized volume in hyperbolic 3-space, motivated by the AdS/CFT holographic principle, giving the audience a view into active research connecting probability, geometry, and physics.