Part I: Around the Center-Focus Problem -- 1 Centers and Limit Cycles -- 2 Darboux Integrability -- 3 Liouvillian Integrability -- 4 Symmetry -- 5 Cherkas’ Systems -- 6 Monodromy -- 7 The Tangential Center-Focus Problem -- 8 Monodromy of Hyperelliptic Abelian Integrals -- 9 Holonomy and the Lotka-Volterra System -- 10 Other Approaches -- Part II: Abelian Integrals and Applications to the Weak Hilbert’s 16th Problem -- 1 Hilbert’s 16th Problem and Its Weak Form -- 2 Abelian Integrals and Limit Cycles -- 3 Estimate of the Number of Zeros of Abelian Integrals -- 4 A Unified Proof of the Weak Hilbert’s 16th Problem for n=2.

This textbook contains the lecture series originally delivered at the "Advanced Course on Limit Cycles of Differential Equations" in the Centre de Recerca Matemàtica Barcelona in 2006. The topics covered are the center-focus problem for polynomial vector fields, and the application of Abelian integrals to limit cycle bifurcations. Both topics are related to Hilbert's sixteenth problem. In particular, the book will be of interest to students and researchers working in the qualitative theory of dynamical systems. This second edition provides updates, further clarifications and remarks, and includes an expanded list of references.