The first part of this chapter is a brief report on the history of quaternionic and Clifford analysis, two research fields that generalize single-variable complex analysis to higher dimension. Appropriate counterparts of the Cauchy–Riemann operator and the Cauchy kernel defined in the frameworks of Hamilton quaternions and Clifford algebras yield spaces of regular and monogenic functions that extend the concept of holomorphic functions. Part 2 introduces Cauchy–Pompeiu and Bochner–Martinelli–Koppelman integral representation formulas with remainders for operator–kernel couples consisting of a first-order differential operator on a Euclidean space with coefficients in a unital Banach algebra and a smooth homogeneous algebra valued kernel. The general formulas underscore the role played by Dirac, Cauchy–Riemann, and Laplace operators in Clifford analysis. The third part of the article is concerned with sharp estimates of fractional integral transforms in a Banach algebra setting and quantitative Hartogs–Rosenthal theorems on uniform approximation of continuos functions on compact sets of Euclidean spaces by solutions of operator–kernel couples, which, in particular, include regular and monogenic functions.