We introduce the Loewner energy for simple planar curves and relate this quantity to ideas and concepts coming from random conformal geometry, geometric function theory, and Teichmuller theory. Many of those concepts arise from areas of mathematical physics. One motivation for the definition of the Loewner energy is that it describes the large deviations of a family of simple random curves called Schramm-Loewner evolutions (SLE) of vanishing parameter. This provides a probabilistic interpretation of the Loewner energy that allows us to establish several interesting geometric properties. We further derive an equivalent characterization of the Loewner energy of Jordan curves using zeta-regularized determinants of Laplacians. This then identifies the Loewner energy with the Kahler potential, introduced by Takhtajan and Teo, of the unique Kahler metric on the Weil-Petersson Teichmuller space.