We address the problem of tracking for a general class of uncertain nonlinear MIMO systems with input quantization, without requiring any matching conditions. We consider the ℒ1 adaptive controller and analyze its performance bounds in the presence of input quantization. We study two common types of quantization, the uniform quantization and the logarithmic quantization. In both cases we provide the transient performance bounds, which are decoupled into two positive terms. One of these terms can be made arbitrarily small by increasing the rate of adaptation, while the other term can be made small by increasing the quantization density. The performance bounds imply that with sufficiently dense quantization and fast adaptation, the output of an uncertain MIMO nonlinear system can follow the desired reference input sufficiently closely. We notice that with ℒ1 adaptive control architecture fast adaptation does not lead to high-gain control and retains guaranteed time-delay margin, which is bounded away from zero. Simulations included in the paper illustrate the results.