One studies the Zariski tangent cone T X π −→ X to an affine variety X and the closure T X of π −1 (Reg(X)) in T X . One focuses on the comparison between T X and T X , giving sufficient conditions on X in order that T X = T X . One considers, in particular, the question of when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Another problem considered here is to understand the impact of the Cohen-Macaulayness or normality of T X on the local structure of X.