This chapter explores the fundamental principles of heat transfer during phase changes, a critical process in various fields like casting, food processing, and ice making. Melting and solidification occur at specific temperatures ( \(T_{f}\) for pure materials) or within a temperature range ( \(T_{s}-T_{l}\) for alloys). Numerical methods tackle complex geometries, and analytical solutions in one dimension offer simple tools for understanding these processes. The chapter focuses on one-dimensional melting/solidification within a domain ( \(\Gamma \) ) containing the metal. This domain is divided by a sharp liquid-solid (L-S) interface with negligible thickness. Analytical solutions for heat transfer in rectangular coordinates are presented, with the moving L-S interface position represented by \(x = s(t)\) . The distinction between solidification in pure metals (occurring at a constant fusion temperature ( \(T_{f}\) ) and alloys (occurring over a temperature range) motivates the chapter’s structure. It lays the groundwork for further exploration of pure metal and alloy solidification of castings with simple and complex shapes.