Links of complex analytic singularities

This is a part survey part research paper studying the local topology of complex analytic spaces. We review and strengthen the results of Kapovich–Koll'ar “Fundamental groups of links of isolated singularities” (1109.4047) and incorporate the paper “Dual graphs of exceptional divisors” (1203.2640) into the current one. A new result is the characterization of the fundamental group of links of Cohen-Macaulay singularities. The last section lists numerous open problems and conjectures. Version 2: Questions 62–65 revised following comments of Shaneson and Maxim. Version 3: Many small changes, especially in section about Questions and Problems.