Classical Hardy spaces, BMO spaces and VMO spaces are associated with the Laplacian operator. Recently, motivated by the Kato conjecture, people began to study spaces (such as Hardy and BMO spaces) associated with other operators. In this lecture, we will introduce VMO spaces associated with two kinds of operators. Firstly, let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb R}^n)$ with suitable upper bounds on its heat kernels, and $L$ has a bounded holomorphic functional calculus on $L^2({\mathbb R}^n)$. We introduce and develop a new function space ${VMO}_L$ associated with the operator $L$. We also prove that a Hardy space $H_{L}^1$, which was introduced by Auscher, Duong and McIntosh, is the dual of our new ${VMO}_{L^{\ast}}$, in which $L^{\ast}$ is the adjoint operator of $L$. Secondly, consider the second order divergence form elliptic operator $L$ with complex bounded coefficients. We also introduce a function space ${VMO}_L$ associated with $L$. Moreover, we prove that the Hardy space $H_{L}^1$, introduced by Hofmann and Mayboroda, is the dual of our ${VMO}_{L^{\ast}}$, in which $L^{\ast}$ is the adjoint operator of $L$. The main tools are theory of tent spaces, functional calculus and Gaffney estimates.