This talk systematically reviews recent advances in differentiable linearization of dynamical systems, a refinement of the classical Hartman-Grobman Theorem. While the theorem guarantees topological conjugacy between a smooth diffeomorphism and its linear part at a hyperbolic fixed point, this conjugacy is generically non-smooth, motivating the core question of its differentiability at the fixed point. We are also concerned with the differentiable normal linearization for partially hyperbolic systems, achieving C^0 conjugacy that is C^1 on the center manifold.