Let $A$ be a matrix with spectrum $\sigma(A)$. The Kreiss Matrix Theorem (KMT), a well-known fact in applied matrix analysis, gives estimates of upper bounds for $\|A^n\|$ if $\sigma(A)$ is in the unit disc, or for $\|\text{e}^{tA}\|$ if $\sigma(A)$ is in the left-half plane based on the resolvent norm, or equivalently the pseudospectra. In this talk, we will discuss the extension of this celebrated theorem to an arbitrary holomorphic function. We will first briefly talk about the norm behavior of matrices that have identical or super-identical pseudospectra. Next we will focus on some generalizations of KMT for holomorphic functions on a general complex domain.