The impacts of the parameters controlling the finite element simulation of instrumented indentation tests and their consequences on numerical results are reviewed in case of homogeneous materials and functionally graded ones. Through a comparative study of the results of different finite element models, the effects of model dimensions (two-dimensional vs. three-dimensional) and its characteristics (boundary conditions, element choice, mesh refinement) on the numerical results are analyzed. Effects associated with rate-independent material behavior, surface roughness, friction, residual stresses and indenter tip bluntness are discussed. It is shown that depending on the material behavior and the maximum indentation depth, the previous conditions can strongly affect the numerical results of indentation tests. The choice of the simulation conditions should be carefully thought as it affects the numerical results. This is also important when building inverse approaches, i.e. methods that extract the material properties using indentation test data, as they are often based on numerical results. Even if the simulation conditions are carefully chosen, some differences may remain between the experimental and numerical results. The last part of this review analyzes the two main sources of remaining differences between the results of finite element simulation and experimentation: the determination of the reference point and the Indentation Size Effect.