This is a survey paper on recent progress in the study of Q-processes with instantaneous state made mainly by the research group led by the first author. After recalling briefly the existence and uniqueness of Q- processes in totally stable, totally instantaneous and mixed cases, some new results are introduced without proof, including the following:
1. For any given Q-matrix of the mixed type, there are infinitely many Q- processes; 2) necessary and sufficient conditions are given for the existence of Q-processes with finitely many stable states and infinitely many instantaneous states; 3) necessay and sufficient conditions are given for the existence of ”B-type” Q-processes with finitely many instantaneous states and infinitely many stable states; 4) necessary and sufficient conditions are given for the existence of Q-processes of a diagonal quasi-Q-matrix with instantaneous states; 5) necessary and sufficient conditions are given for the uniqueness and existence of birth-and-death Q-processes with finite instantaneous state. All the birth-and-death Q-processes are constructed.