<p>The Su-Schrieffer-Heeger (SSH) model is a foundational framework in the study of one-dimensional topological insulators, offering significant insights into condensed matter physics and quantum systems. Originally formulated to describe polyacetylene, a one-dimensional polymer, the SSH model effectively captures the essence of electron-phonon interactions and topological phases. Its simplicity and ability to illustrate phenomena such as edge states and topologically protected modes-robust against perturbations-make it an invaluable tool for exploring more complex systems. Applications range from quantum computing to the design of novel materials with topological properties. Additionally, the SSH model serves as a critical theoretical framework for investigating symmetry-protected topological phases, providing deeper understanding of non-trivial band structures and their relevance to quantum technologies. In this article, we comprehensively examine the SSH model, covering fundamental aspects such as the Peierls distortion, electronic band structure, the mathematical foundations of the winding number, edge states, and its connection to modern polarization theory and the inversion-symmetric Rice-Mele model. Additionally, we present new analytical insights into the SSH model’s response functions, including optical conductivity and specific heat with an updated overview of recent works and developments on this model. These in-depth analysis, explanations and observations will help readers across all genres and will contribute to open intriguing avenues for further research in topological quantum matter.</p>