Totally positive matrices are characterized by having all their minors positive. They appear in various areas of physics and mathematics, including oscillations in mechanical systems, quantum groups, and algebraic geometry. It has been known since the work of Fomin that the two-point correlations functions of the two-dimensional Gaussian free field satisfy total positivity. I will present an analogous result for the correlations of the planar Ising model. The idea is to prove that determinants of such correlations have interpretations in terms of probabilities of events in the random current model. A natural open question is to identify all planar totally positive spin systems.