Abstract: To understand the dynamics of the Ginzburg-Landau vortices, the Ginzburg-Landau equation without the magnetic field has been studied. There are two important solutions. One is the single-vortex solution with degree one, and the other is the multi-vortex solution. For the single-vortex solution, the small perturbation of the symmetric vortex solution has been considered, and the spectrum of the linearized operator $L_\epsilon$ is crucial to know the dynamic stability of the single-vortex solution. The main result is about the estimates of the principal eigenvalues. For the multi-vortex solution, the main result is to solve the conjecture in the book of F. Bethuel, H. Brezis and F. Helein : Given a nondegenerate critical point of the renormalized energy, there is a sequence of solutions $u_\epsilon$ of the Ginzburg-Landau equations such that $u_\epsilon$ converge (up to a subsequence) to a canonical harmonic map as $\epsilon$ tends to zero.