The relationships among various quantum automata are clarified, and in particular, variously equivalent characterizations of quantum automata are established. The G-quantum automata, g-quantum automata, (generalized) quantum automata, G-quantum grammars and g-quantum grammars are presented, and their connections to some of the other existing quantum automata are expounded. Under a certain condition, the equivalence between the G(g)-quantum automata and G(g)-quantum grammars is discussed. Therefore the problem of quantum grammars generating quantum regular languages is solved. The relation between quantum and regular languages is dealt with, and particularly, two open problems proposed by Gudder are answered. Finally, the method of decreasing the dimensions of spaces of states on quantum automata is given.