The study of real planar curves dates back to antiquity, where the ancient Greeks studied curves defined on the plane cut out by polynomials of two variables. We’ll provide a friendly overview to beautiful formulas of Plücker which govern the “shape” of planar curves. We will discuss the Shapiro—Shapiro conjecture and connections to the real Schubert calculus, and end by presenting some new conjectures and computational evidence joint with Frank Sottile.