It is known that geodesics are distance-minimizing curves. A conformal structure on a manifold is a class of metrics given up to a factor. Consequently, the notion of geodesics becomes meaningless on conformal manifolds. However, there is a conformally invariant equation of the third order whose solutions constitute a well-defined class of curves. In the flat case, these are circles. In general, they are called conformal geodesics. Since the equation is of the third order, it is unclear how to interpret it in terms of a variational principle (the Euler-Lagrange equations are of even order). In this talk I will show how to overcome these difficulties. I will also discuss a variational formulation of the Schwarzian equation.