This paper presents the L1 adaptive controller for a class of uncertain systems in the presence of time and state-dependent unknown nonlinearities. The L1 controller achieves performance specifications, defined via a time-varying reference system, using output-feedback. 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 reducing the sample time. Simulations verify the theoretical findings.