The theory of primitive forms was introduced by Kyoji Saito in early 1980s, which was first known in singularity theory and has attracted much attention in mirror symmetry recently. In this talk, we will introduce a complex differential geometric approach to primitive forms, using compactly supported polyvector fields. We will first introduce the notion of primitive forms, making it acceptable to general audience. We will also introduce an explicit perturbative construction of primitive forms with respect to opposite filtrations and primitive elements. Depending on the time, we will illustrate the perturbative formula by various examples, which may include some old examples of simple elliptic singularities and the mirror Laudau-Ginzberg model of P^1 and new example of exceptional unimodular singularity of type E12. This is my recent joint work with Si Li and Kyoji Saito.