I Introduction -- I.1 A paradigm -- I.2 Experimental results -- I.3 Modeling for theoretical analysis -- I.4 Arrangements of topics in the text -- II Statement of the Problem and Basic Tools -- II.1 Nondimensionalization, parameters -- II.2 Functional frame and basic properties -- II.3 Linear stability analysis -- II.4 Center Manifold Theorem -- III Taylor Vortices, Spirals and Ribbons -- III.1 Taylor vortex flow -- III.2 Spirals and ribbons -- III.3 Higher codimension bifurcations -- IV Mode Interactions -- IV.1 Interaction between an axisymmetric and a nonaxisymmetric mode -- IV.2 Interaction between two nonaxisymmetric modes -- V Imperfections on Primary Bifurcations -- V.1 General setting when the geometry of boundaries is perturbed -- V.2 Eccentric cylinders -- V.3 Little additional flux -- V.4 Periodic modulation of the shape of cylinders in the axial direction -- V.5 Time-periodic perturbation -- VI Bifurcation from Group Orbits of Solutions -- VI.1 Center manifold for group orbits -- VI.2 Bifurcation from the Taylor vortex flow -- VI.3 Bifurcation from the spirals -- VI.4 Bifurcation from ribbons -- VI.5 Bifurcation from wavy vortices, modulated wavy vortices -- VI.6 Codimension-two bifurcations from Taylor vortex flow -- VII Large-scale EfTects -- VII. 1 Steady solutions in an infinite cylinder -- VII.2 Time-periodic solutions in an infinite cylinder -- VII.3 Ginzburg-Landau equation -- VIII Small Gap Approximation -- VIII.1 Introduction -- VIII.2 Choice of scales and limiting system -- VIII.3 Linear stability analysis -- VIII.4 Ginzburg-Landau equations.