<p>This article serves as an expository and introductory work, exploring generalized Fibonacci numbers through the lens of integer compositions and graph theory. It is particularly suited for students interested in discrete mathematics, providing a combination of combinatorial arguments, bijective proofs, and connections to various discrete structures. Key topics include the relationship between Fibonacci numbers and compositions, graphical representations such as bargraphs and Young diagrams, and extensions to tribonacci and tetranacci sequences. Additionally, the article introduces colored compositions and towers of compositions.</p><p>To engage readers further, a set of problems is included, designed to be approachable for students and to serve as an entry point for those intrigued by the topics discussed. These problems aim to foster a deeper understanding and appreciation of the rich interplay between Fibonacci sequences and combinatorics.</p>