This chapter discusses the properties of holomorphic functions in several complex variables, focusing on plurisubharmonic functions. Holomorphic functions are complex-differentiable functions in a domain $ D \subset \mathbb{C}^n $, satisfying the Cauchy-Riemann equations. The "catch-all" condition of continuous differentiability can be weakened, as shown by the Hartogs theorem, which states that if a function is analytic in each variable separately, it is jointly analytic. The space of all holomorphic functions in $ D $, denoted $ \mathcal{H}(D) $, is a linear subspace of the space of continuous functions $ \mathcal{C}(D) $. It is endowed with the topology induced by seminorms defined on compact subsets of $ D $, or more generally, $ \mathcal{H}(D) $-bounded subsets. These subsets are those on which all holomorphic functions are bounded. The topology of $ \mathcal{H}(D) $ is equivalent whether defined by compact subsets or $ \mathcal{H}(D) $-bounded subsets. The chapter also discusses the integral representation of holomorphic functions in polycircular domains. A polycircular domain is a domain in $ \mathbb{C}^n $ defined by a center and a polyradius. Any holomorphic function in a domain containing a polycircular domain can be represented by an n-tuple Cauchy integral. The chapter also proves that any holomorphic function is infinitely differentiable, and provides an estimate for the derivatives of such functions. Finally, it shows that $ \mathcal{H}(D) $ is a closed subspace of $ \mathcal{C}(D) $ and $ \mathcal{E}(D) $, making it a Fréchet space. The LCS structure on $ \mathcal{H}(D) $ is the same whether induced from $ \mathcal{C}(D) $ or $ \mathcal{E}(D) $.This chapter discusses the properties of holomorphic functions in several complex variables, focusing on plurisubharmonic functions. Holomorphic functions are complex-differentiable functions in a domain $ D \subset \mathbb{C}^n $, satisfying the Cauchy-Riemann equations. The "catch-all" condition of continuous differentiability can be weakened, as shown by the Hartogs theorem, which states that if a function is analytic in each variable separately, it is jointly analytic. The space of all holomorphic functions in $ D $, denoted $ \mathcal{H}(D) $, is a linear subspace of the space of continuous functions $ \mathcal{C}(D) $. It is endowed with the topology induced by seminorms defined on compact subsets of $ D $, or more generally, $ \mathcal{H}(D) $-bounded subsets. These subsets are those on which all holomorphic functions are bounded. The topology of $ \mathcal{H}(D) $ is equivalent whether defined by compact subsets or $ \mathcal{H}(D) $-bounded subsets. The chapter also discusses the integral representation of holomorphic functions in polycircular domains. A polycircular domain is a domain in $ \mathbb{C}^n $ defined by a center and a polyradius. Any holomorphic function in a domain containing a polycircular domain can be represented by an n-tuple Cauchy integral. The chapter also proves that any holomorphic function is infinitely differentiable, and provides an estimate for the derivatives of such functions. Finally, it shows that $ \mathcal{H}(D) $ is a closed subspace of $ \mathcal{C}(D) $ and $ \mathcal{E}(D) $, making it a Fréchet space. The LCS structure on $ \mathcal{H}(D) $ is the same whether induced from $ \mathcal{C}(D) $ or $ \mathcal{E}(D) $.