in their celebrated work, p. li and s.-t. yau proved the famous li-yau gradient bound for positive solutions of the heat equation on manifolds with ricci curvature bounded from below. since then, li-yau type gradient bounds has been widely used in geometric analysis, and become a powerful tool in deriving geometric and topological properties of manifolds. in this talk, we will present our recent works on li-yau type gradient bounds for positive solutions of the heat equation on complete manifolds with certain integral curvature bounds, namely, $\vert ric_\vert$ in $l^p$ for some $p>n/2$ or certain kato type of norm of $\vert ric_\vert$ being bounded together with a gaussian upper bound of the heat kernel. these assumptions allow the lower bound of the ricci curvature to tend to negative infinity, which is weaker than the assumptions in the known results on li-yau bounds. these are joint works with qi s. zhang.