The phenomena of synchronization can be easily found in a variety of natural systems. The first reported observation of synchronization is one Dutch scientist’s discovery: Christiaan Huygens realized that two pendulum clocks hanging on the wall have always ended up swinging in exactly the opposite direction from each other in 1665. Since then, people have recognized synchronization phenomena in various areas including circadian rhythms, electrical generators, Josephson junction arrays, intestinal muscles, menstrual cycles, and fire flies. Although it is studied in many different scientific disciplines such as applied mathematics, biology and nonlinear dynamics, the underlying mechanism of synchronization has remained a mystery. Among a number of mathematical models, the differential equations proposed by Kuramoto and Winfree have received considerable attention. In this lecture series, we mostly focus on the analysis for Kuramoto system.