This talk is concerning gradient estimates and its applications on graphs. Li-Yau inequality is a powerful tool for studying positive solutions to the heat equation. By studying the heat semigroup, we derive Li-Yau inequality on non-negatively curved graphs. From it, we obtain some applications of Li-Yau inequality, such as Harnack inequality, Guassian heat kernel bounds. Furthermore, we get some gradient estimates of positive solutions to some differential inequalities. This talk is based on some joint works with Chao Gong, Pual Horn, Yong Lin, Yunyan Yang and Shing-Tung Yau.