Orthogonal polynomials on quadratic curves in the plane are studied.These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and twolines. For an integral with respect to an appropriate weight function defined onany quadratic curve, an explicit basis of orthogonal polynomials is constructedin terms of two families of orthogonal polynomials in one variable. Convergenceof the Fourier orthogonal expansions is also studied in each case. We discussapplications to the Fourier extension problem, interpolation of functions with sin-gularities or near singularities, and the solution of Schr ̈odinger’s equation withnon-differentiable or nearly-non-differentiable potentials.