Melting of elemental solids can be identified and appreciated as a particularly simple example of discontinuous phase transitions involving condensed phases. Motivated, on the one hand, by the need to improve the microstructural quality of laser-crystallized columnar-grained polycrystalline Si films for manufacturing advanced AMOLED displays and, on the other hand, to investigate the fundamental details associated with phase transformations transpiring in condensed systems, this thesis examines the initiation and evolution of melting in polycrystalline thin films. Distilling the essence of the classical nucleation theory and extending its description to address more general cases of phase initiation and evolution, a general thermodynamic method based on capillarity effect is developed and applied to determine the shape of solid/liquid interfaces that are in mechanical equilibrium. We first explicitly identify and build our analysis based on how the shape of solid/liquid interfaces must comply with the contact angle conditions at the junctions and also the property of constant mean curvature. Bi-crystal and tri-crystal models are presented to capture the microstructural features such as junctions and vertices of interfaces in polycrystalline thin films. At each of the potential melt initiating sites, the parameter space of contact angles is divided into domains depending on the shape of the solid/liquid interface that can be established in mechanical equilibrium. Melting initiation mechanisms are subsequently determined based on the permissible shape for each domain. This analysis is further extended to the edges and corners of embedded cubic crystals (with nonidentical contact angles at different faces). Secondly, in order to facilitate the thermodynamic analysis of the melting initiation and interface propagation, we extend our curvature-evolution-centric method to identify and develop what we consider as the central function for discontinuous phase transitions. Specifically, starting with a local governing condition, identifies and builds on two curvatures: ρ^E (𝑉) and ρ* (𝑇). ρ^E (𝑉) captures the evolution of the mean curvature of the solid/liquid interface as a function of liquid volume for the case in which the mechanical equilibrium condition is satisfied, whereas ρ* (𝑇) incorporates the temperature effect on the difference between the volumetric free energy of solid and liquid phases using the corresponding equilibrium mean curvature. We define and identify the interface driving stress function ƒ(𝑉,𝑇)=∂𝐺/∂𝑉=σ(ρ^E (𝑉)-ρ* (𝑇)) of the phase transition as being an important fundamental quantity, which can be directly derived by taking the difference of the two curvature terms. In contrast to the conventional analysis that requires integration of volumetric and interfacial free energy terms over various geometric domains to derive the total free energy as a function of volume for a given temperature, this formation completely disentangles geometry from the thermodynamic aspects of the phase transition and allows them to be treated separately. In addition to providing essentially all relevant thermodynamic information about the phase initiation and evolution, the above method readily permits the use of powerful general-purpose numerical tools to calculate the potentially complex geometry of the solid/liquid and other interfaces and obtain ρ^E (𝑉) directly as the output. Plotting the ρ^E (𝑉) function together with the temperature-dependent iso-curvature line, ρ* (𝑇), unveils the critical thermodynamic information regarding the melting transition at the temperature, such as whether equilibrium points exist, the number of equilibrium points, their stability, and their corresponding volumes. The change of free energy as a function of liquid volume can be derived through integration of the interface driving stress function. The velocity of the solid/liquid interface is simply proportional to the interface driving stress function. The application of this method is demonstrated in both shape-preserving (which we term as isomorphic) and shape-changing (which we term as non-isomorphic) examples. The analysis and findings presented in this thesis are relevant and useful for understanding discontinuous phase transitions, in general, and can be particularly so for small, confined, and embedded systems that are increasingly being utilized in modern technologies.