Encounter with character sums

The theory of Dirichlet characters is an important but extremely difficult subject in number theory. Almost 100 years have passed since the fundamental contribution of G. Pólya and I. M. Vinogradov (1918), there were very few substantial progress like D. A. Burgess's work in late 1950s. A. A. Karatsuba created a new method to deal with character sums in late 1960s, from which many an arithmetical applications have been derived. The past decades have seen some promising changes due to the introduce of new ideas and tools. In this talk I will present some old and new results in the field of character sums, then revisit several open problems related to Dirichlet characters. Finally, I will present my recent joint works with C. H. Jia and M. A. Korolev.