It is known that the adiabatic approximation in time-dependent density functional theory usually provides a good description of low-lying excitations of molecules. In the present work, the capability of the adiabatic nonempirical meta–generalized gradient approximation (meta-GGA) of Tao, Perdew, Staroverov, and Scuseria (TPSS) to describe atomic and molecular excitations is tested. The adiabatic (one-parameter) hybrid version of the TPSS meta-GGA and the adiabatic GGA of Perdew, Burke, and Ernzerhof (PBE) are also included in the test. The results are compared to experiments and to those obtained with two well-established hybrid functionals PBE0 and B3LYP. Calculations show that both adiabatic TPSS and TPSSh functionals produce excitation energies in fairly good agreement with experiments, and improve upon the adiabatic local spin density approximation and, in particular, the adiabatic PBE GGA. This further confirms that TPSS is indeed a reliable nonhybrid universal functional which can serve as the starting point from which higher-level approximations can be constructed. The systematic underestimate of the low-lying vertical excitation energies of molecules with time-dependent density functionals within the adiabatic approximation suggests that further improvement can be made with nonadiabatic corrections.