Abstract: Grover’s algorithm has attracted much attention ever since it was proposed, because it has a quadratic speedup over classical algorithm for searching unstructured database. However, the original Grover’s algorithm usually cannot obtain the target elements with certainty, even if the proportion of target elements is known. To this end, exact Grover’s quantum search algorithms were proposed as extensions of the original Grover’s algorithm, which can output the target element with certainty, while maintaining the quadratic speedup. This paper systematically sorts out the three existing exact Grover’s quantum search algorithms, introducing in detail the algorithm process, parameter settings, and the geometric intuition behind them. Moreover, the lower bound on the query complexity of these algorithms is shown, under both situations when the proportion of target elements is known or unknown.