The heat kernel provides an important link between probability theory and partial differential equation. In probability theory, the heat kernel of an operator $L$ is the transition density $p(t,x,y)$ (if it exists) of the Markov process $X$, which possesses $L$ as its infinitesimal generator. In the field of of partial differential equation, it is called the fundamental solution of the heat equation. However, except in a few special cases, obtaining an explicit expression of $p(t, x, y)$ is usually impossible. That`s why finding sharp estimates of $p(t, x, y)$ is a fundamental issue both in probability theory and partial differential equation. This talk consists of three parts. In the first section, we discuss heat kernel estimates for symmetric Dirichlet form corresponds to Hunt processes. Next we will see the heat kernel estimates for non-symmetric nonlocal operators concern with jump processes. The last part of talk is devoted to the applications of heat kernel estimates. In particular, we deals with the boundary regularity estimates for nonlocal operators with kernels of variable orders. Then, I will introduce the laws of iterated logarithms for Markov processes, which is my ongoing research project. Heat kernel and its estimates plays an important role in both problems.