The common Gronwall inequalities are divided into integral and differential forms. Firstly, for the common integral Gronwall inequalities, the purpose is to give a new proof method, which is different from the proof method of exponential function multiplied by the two ends of former inequalities, but is proved by using the basic integral formula, and the uniqueness of the solutions of the ordinary differential equations of the first order type is proved by using this inequality. Secondly, the purpose is to generalize differential Gronwall inequality, and the uniqueness of the solutions of wave equation and the estimation of the solution energy of heat conduction equation are proved by using the basic differential equalities. Furthermore, by using variable substitution, derivation formula and the basic differential Gronwall inequality, the Gronwall inequality of the first order differential type is generalized into two cases: the rightend control term rises from 1 to the αth power (α>0); the Gronwall inequality of the first order differential type is extended to the Gronwall inequality of the second order differential type, and the conclusion similar to that of first order differential type is obtained.