This paper delves into the intricate "fine structure" of non-standard models of Peano Arithmetic (PA). While the standard model of natural numbers is unique up to isomorphism, the compactness theorem guarantees the existence of uncountably many non-isomorphic non-standard models. We explore their characteristic order types, which uniformly begin with an initial segment isomorphic to the natural numbers followed by dense arrangements of copies of the integers. The study focuses on key model-theoretic properties such as recursive saturation, resplendency, and the role of indiscernibles in shaping the internal complexity of these models. Tennenbaum's Theorem, demonstrating the non-recursiveness of addition and multiplication in any countable non-standard model, highlights a fundamental distinction from the standard model. We examine the implications of these structural insights for foundational questions in mathematics, including definability, interpretability, and the limits of formal axiomatization, offering a comprehensive overview of current understanding and outlining avenues for future research into these fascinating arithmetical universes.