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Dynamics of polynomials in the complex plane is well known. In higher dimensions, the tools of complex (pluri)potential theory play an important role in the study of complex dynamics. We will first discuss dynamics of the polynomial automorphisms of $\mathbb{C}^2$, namely, Hénon maps. For these maps, we can construct Fatou sets, Julia sets and invariant probability measures which are analogous to those in the complex plane. In $\mathbb{C}^3$, there is a nice classifi- cation of polynomial automorphism up to affine conjugates. Some of these classes were extensively studied in terms of their dynamical properties. We will mention the recent results on the dynamics of one of the automorphism class in $\mathbb{C}^3$.