This textbook presents efficient analytical techniques in the local bifurcation theory of vector fields. It is centred on the theory of normal forms and its applications, including interactions with symmetries. Attention is given to examples with reversible vector fields, including the physical example given by the water waves. The second part deals with the Couette-Taylor hydrodynamical stability problem, between concentric rotating cylinders. The spatial structure of various steady or unsteady solutions is covered. The text also studies bifurcations from group orbits of solutions in an elementary way. The third part analyzes bifurcations from time periodic solutions of autonomous vector fields. A normal form theory is developed, covering all cases, and emphasizing a partial Floquet reduction theory, which is applicable in infinite dimensions. Studies of period doubling as well as Arnold’s resonance tongues are included in this part.