Riemann surfaces arise naturally when one encounters the multiple-valued behavior of the complex logarithm or square root. In the preceding chapters, we have learned how to take a single-valued branch of such a function. In this chapter, we will introduce a new concept of dealing with multiple-valuedness by using analytic continuation of holomorphic functions, which leads naturally to the notion of Riemann surfaces. A systematic treatment of analytic continuation on Riemann surfaces is given in Appendix G. Here we will only demonstrate many explicit examples.