This thesis is divided into two parts. In the first one the creation of a complete numerical tool, from the mesh generation to the data treatment, is presented. However time consuming this part has been, this is not the most important one. The second and more important part is concerned with the description and physical interpretation of the results obtained with the numerical tool previously developed. The numerical tool, which exists in its quasi-final version since mid-1997, fully satisfies the following requirements. It is specifically designed to study the flow in the annular space between coaxial, differentially-rotating cylinders of finite length. The spectral element method is used for the space discretization. The study of transition requires the high accuracy warranted by this type of method. The numerical scheme has also to be efficient. This requirement is satisfied, due to the fully-explicit time scheme adopted, where both diffusive and non-linear terms of the Navier-Stokes equations are treated explicitly, and the direct inversion of the pseudo-Laplacian matrix applied to the pressure. This inversion is performed in the most efficient way with a fast diagonalization technique. In the Reynolds numbers range we are interested in, the time-step limitation due to the linear viscous term is only slightly more stringent than the one due to the non-linear term. The last requirement that has been fulfilled has been to design as simple a code as possible. The entire code is constructed from a number of well-known algorithms, fitted together to enhance efficiency. However, the way we regularize the boundary conditions is new. It represents the physics more precisely than Tavener et al. [65]. There is a second original feature in our code, which is linked to the time scheme. We derived our time discretization from the scheme of Gavrilakis et al. [31]. Their scheme cannot be applied to cylindrical coordinates as it is. We thus had to modify it. The second part of the thesis consists of the numerical study of the first transitions of the Taylor-Couette flow in a finite-length geometry. The aspect ratio between the length of the cylinders and the gap between them has been chosen equal to twelve; this is small enough for the effects of the upper and lower boundaries of the flow to be significant. It is believed that these end-effects play a non-negligible role in the transition of the flow. On the other hand, the aspect ratio is large enough to make qualitative comparisons with the infinite-length case, which has been studied extensively both theoretically and numerically. The transition process depends on a relatively large number of parameters. Our investigation focuses on the case where the cylinders rotate in opposite directions. The study of counter-rotating Taylor-Couette flow for a large but finite aspect ratio is the main originality of this thesis. Furthermore, the physical mechanism of the appearance of the second bifurcation in classical Taylor-Couette flow, where only the inner cylinder is rotating, is described in the light of the non-linear interaction between velocity and vorticity.