This paper is an expository self-contained article on results concerning the construction of a vector-valued holomorphic functional calculus recently obtained by the author and by \textit{R. E. Harte} and \textit{C. Taylor} as coauthors in a series of three papers [Math. Proc. R. Ir. Acad. 101A, No. 2, 177--196 (2001; Zbl 1028.46076), ibid. 197--220 (2001; Zbl 1028.46011), ibid. 103A, No. 1, 61--92 (2003; Zbl 1060.46031)]. Let \(\mathcal A\) be a commutative unital Banach algebra, let \(X\) be a Banach space, let \(\gamma\) be any uniform crossnorm on \({\mathcal A} \otimes X\) and let \({\mathcal A}\widehat{\otimes}_\gamma X\) be the associated tensor product. For elements \({\mathbf a} \in {\mathcal A}\widehat{\otimes}_\gamma X\), the Waelbroeck spectrum is denoted by \(\sigma({\mathbf a})\). The author constructs a holomorphic functional calculus \[ \Theta_{\mathbf a}:({\mathcal H}(\sigma({\mathbf a})),\tau_0) \rightarrow {\mathcal A} \] (\(\tau_0=\) compact open topology) provided that \(X\) has the so-called \(({\mathcal A},\gamma)\)-extension property. This property can briefly be defined as follows: For any \(n\)-homogeneous polynomial \(P \in {\mathcal P}(^nX)\), the assignment \(P_{\mathcal A}(a \otimes x)=a^nP(x)\) extends (uniquely) to an \(n\)-homogeneous polynomial \(P_{\mathcal A} \in {\mathcal P}(^n({\mathcal A}\widehat{\otimes}_\gamma X),{\mathcal A}\widehat{\otimes}_\gamma X)\) satisfying the boundedness condition \(\| P_{\mathcal A}\| \leq c^n\| P\| \) for a certain \(c>0\) independently of \(P\) and \(n\). For \(\gamma = \pi\), the projective topology, each Banach space \(X\) has the \(({\mathcal A},\pi)\)-extension property.