The stability and instability of equilibrium points of nonlinear autonomous fractional differential systems in finite-dimensional Euclidean spaces are investigated. The analysis is carried out by means of the linearization method. A distinctive feature compared with the classical case of ordinary differential equations is the construction of a Lyapunov–Perron operator together with a weighted norm adapted to the fractional framework. In addition to establishing the local convergence of solutions to the equilibrium, the rate of this convergence is also determined.