The lattice Boltzmann method (LBM) has become an effective numerical technique for computational fluid dynamics (CFD) in recent years. It has many advantages over the conventional computational methods like finite element and finite difference methods. The method is characterised by simplicity, easy treatment of boundary conditions and parallel feature in programming that makes it ideal for solving large-scale real-life problems. This thesis presents the development and applications of a lattice Boltzmann model for both steady and unsteady two-dimensional axisymmetric flows. The axisymmetric flows are described by three-dimensional (3D) Navier-Stokes equations, which can be solved by three-dimensional (3D) lattice Boltzmann method. If cylindrical coordinates are applied, such 3D equations become 2D axisymmetric flow equations. However, they cannot be solved using the 2D standard LBM. In order to study more complicated axisymmetric flow problems by 2D LBM, in this thesis, firstly, the revised axisymmetric lattice Boltzmann D2Q9 model (AxLAB®) is applied and tested for some benchmark for axisymmetric laminar flows and more complicated flows including 3D Womersley flow and forced axisymmetric cold-flow jets, and flows with swirl such as the cylindrical cavity flows and the swirling flow in a closed cylinder with rotating top and bottom. Secondly, the AxLAB® is extended to simulate turbulent flows and non-Newtonian fluid flows. A well-known power-law scheme is incorporated into the AxLAB® to simulate the non-Newtonian fluid flow: the Taylor Couette flows for Newtonian and non-Newtonian fluids are simulated and compared. The combined effects of the Reynolds number, the radius ratio, and the power-law index on the flow characteristics are analysed and compared with other literatures. All the numerical results are also compared with the existing numerical results or experimental data reported in the literature to demonstrate the accuracy of the model. Thirdly, a further developed AxLAB® is presented to simulate the turbulent flows. The turbulent flow is efficiently and naturally simulated through incorporation of the standard subgrid-scale stress (SGS) model into the axisymmetric lattice Boltzmann equation in a consistent manner with the lattice gas dynamics. The model is verified by applying it to several typical cases in engineering: (i) pipe flow through an abrupt axisymmetric constriction, (ii) axisymmetric separated and reattached flow and (iii) pulsatile flows in a stenotic vessel. All the numerical results obtained using the present methods are compared with experimental data and other available numerical solutions, indicating good agreements. This shows that the improved AxLAB® is simple and is able to predict axisymmetric turbulent, non-Newtonian complicated flows at good accuracy.