Abstract: Let Q=\left(\beginarraycccc-\infty & b_1 & b_2 & \cdots \\ f_1 & -q_1 & & \\ f_2 & & -q_2 & \\ : & & & \ddots\endarray\right),where 0≤q_i,b_i,f_i＜∞,q_i≥f_i,（i∈I）, Such matrix Qis calleda generalized Kolmogorov matrix. Corresponding to this matrix, the criteria for the existence or uniqueness of Q-processes are given. By using it, some necessary and sufficient conditions for Q to be a Q-matrix or an honest Q-matrix are given respectively. These results improve those obtained in 2-7.