Natural circulation driven nuclear reactors are prone to flow instability during the startup transients. This paper intends to provide the state-of-the-art reviews on the theoretical analysis and experimental studies on flow instability in three types of natural circulation driven reactors, ranging from conventional nuclear reactors to small modular reactors. Brief overviews of three categories of startup flow instability, i.e., density wave oscillations, flashing instability, and Geysering instability, are provided. A critical review is conducted for the scaling analysis and design of small scaled test facility. The review of obtaining quasi-steady state stability maps in the dimensionless stability plane through frequency domain analysis and experimental tests provides the state-of-the-art methodology of analyzing the flow instability. Experimental startup instability during different initial startup procedures is reviewed. Although extensive efforts have been made to study the flow instability, further work is required to improve the scaling ability of experimental investigation and the accuracy of code analysis. Some discussions for future research directions are given.