This paper deals with the design, performance, and robustness analysis of nonlinear attitude complementary filters on SO(3). We derive explicit time solutions of the attitude estimation error dynamics of the filters (in the disturbance-free case) and analyze their performance and robustness with respect to measurement disturbances. The stability and performance properties of the filters can be easily deduced from the obtained closed-form solutions. A new class of attitude complementary filters on SO(3) with state-dependent gains is proposed and shown to exhibit improved performance and robustness properties compared to the fixed-gain traditional complementary filter on SO(3).