We start by exploring and categorizing the literature related to FOs. The results of Scopus queries are graphically presented in Figure . Nearly 10,000 papers have been published mentioning the experimental observation of FO, with 6,000 focusing on theoretical approaches and 3,000 employing computational methods. . Upon thorough examination, the cited works could be divided into areas: Scanning Transmission Microscopy (STM), Green's function (GF), Linear Response Theory (LR), and Surface States (SS). In this review, we structure and categorize these research findings, consolidate similar results, and offer clear, step-bystep derivations of theoretical advancements using a unified, coherent and concise mathematical approach. We have also cataloged noteworthy experimental papers and highlighted peculiarities such as the observation of asymmetric anisotropic Friedel oscillations in experiments, along with derivations of FOs in d-dimensional cases. Most importantly, we aim to under-stand whether indirect interactions occurring in metallic systems, graphene, or any electron gas system can be explained through a detailed analysis of Friedel oscillations (please see figure ).