This paper presents the L₁ adaptive controller for a class of uncertain nonlinear systems in the presence of time and state dependent unknown nonlinearities with unmodeled nonlinear dynamics. The L₁ controller achieves performance specifications, defined via a time-varying reference system. We prove that subject to standard regularity assumptions, the L₁ adaptive controller ensures uniformly bounded transient and steady-state tracking for system's input and output signals simultaneously. The performance bounds can be systematically improved by increasing the rate of adaptation. Simulations of a nonlinear system, coupled to a Lorenz attractor, verify the theoretical findings.