We propose a novel optimization algorithm for continuous functions using geodesics and contours under conformal mapping.The algorithm can find multiple optima by first following a geodesic curve to a local optimum then traveling to the next search area by following a contour curve. To improve the efficiency, Newton-Raphson algorithm is also employed in local search steps. A proposed jumping mechanism based on realized geodesics enables the algorithm to jump to a nearby region and consequently avoid trapping in local optima. Conformal mapping is used to resolve numerical instability associated with solving the classical geodesic equations. Geodesic flows under conformal mapping are constructed numerically by using local quadratic approximation. The parameters in the algorithm are adaptively chosen to reflect local geometric features of the objective function. Comparisons with many commonly used optimization algorithms including gradient, trust region, genetic algorithm and global search methods have shown that the proposed algorithm outperforms most widely used methods in almost all test cases with only a couple of exceptions.