Numerical methods in fluid dynamics and heat transfer are experiencing a remarkable growth in terms of the number of both courses offered at universities and active researches in the field. There are some software packages available that solve fluid flow problems. Nevertheless, Computational Fluid Dynamics (CFD) codes are progressively being accepted as design tools by the industry. Nowadays users of CFD need to be fairly knowledgeable, which requires instruction of both students and working engineers. The present text is a starting point to immerse the student in the tissues of the field. The two main objectives of this project are: to acquire a basic training in the numerical resolution of the governing equations in the heat transfer and fluid dynamics, and to get used to CFD and Heat Transfer (HT) codes and acquire the skills to critically judge their quality, this is, apply code verification techniques, validation of the used mathematical formulations and verification of numerical solutions. In the present text, fundamental methods for solving the transport phenomena are covered. Chapter 1. ‘Discretization and solvers’ contains the fundamental numerical method since the physical phenomena must be described through appropriate differential equations. Chapter 2. ‘Heat conduction methods’ is the construction base of the numerical method, therefore emphasis on concepts and calculation details are given here. Chapter 3. ‘Analysis of the general convection-diffusion equation’ is focused on the interaction of convection and diffusion, with the flow field known in advance. Finally, the calculation of the velocity field itself is treated in Chapter 4. ‘Incompressible flow method using the Navier-Stokes equations’. This chapter represents an effort to employ the Fractional Step Method (FSM) in the solution of the Navier-Stokes equations with the aim to obtain solutions for diverse Reynolds numbers and mesh refinements. The problems presented and solved are intended to be a material base over which analysis, discussion and conclusions are developed. The Smith-Hutton problem is addressed since many of the features commonly encountered in practical convection-diffusion problems are here present. Different numerical schemes are submitted and their pros and cons are described. Moreover, the robustness of the Fractional-Step Method (FSM) has been demonstrated using the Driven cavity flow benchmark problem. Detailed accurate results have been presented for this problem. Up to 128x128 computational points and Reynolds as high as 3200 have been considered. Keywords – numerical methods, fluid dynamics, heat and mass transfer, convection-diffusion, convective schemes, Smith-Hutton, incompressible flow, Navier-Stokes, fractional-step method, staggered meshes, Driven cavity flow.