Three-degree-of-freedom nonlinear vibratory systems with 1: 1: 1 internal resonance and with cubic nonlinearity are considered. Attention is focused on the responses of the systems near a primary resonance point. The systems are assumed to be subjected to a harmonic excitation on one of their three modes. The systems can be divided into two groups depending on the types of responses. In one group both single-mode and multi-mode responses are possible and in the other only multi-mode response is possible. Typical systems belonging to the former and the latter group are studied. The characteristics of their responses are discussed, and the occurrence of dynamic phenomena such as chaotic vibrations with different routes is shown. Among the phenomena some have not previously been reported on in relation to studies of two-degree-of-freedom vibratory systems with 1: 1 internal resonance.