In recent years, discrete complex analysis has experienced significant advancements, despite its more recent development compared to the well established field of continuous complex analysis. While continuous complex analysis has a long history, its discrete counterpart is a newer and actively evolving area of study. This talk will begin with an overview of various discretizations in complex analysis. We will then focus on a linear theory of discrete complex analysis on planar quad-graphs, utilizing its medial graph. In this context, we present discrete analogs of fundamental objects in complex analysis and demonstrate how "linear" statements and proofs from continuous complex analysis can be directly translated to the discrete setting. Specifically, we will explore discrete versions of key theorems, including Green's first and second identities, as well as Cauchy's integral formulas for holomorphic functions and their derivatives. Although the asymptotic behavior of discrete Green's functions and Cauchy's kernels is generally unknown, we will discuss their asymptotics on parallelogram graphs. This presentation is based on joint work with Alexander Bobenko.