Nucleation is a phenomenon of broad scientific interest and technological importance. It refers to the very early stages of the formation of a new phase, which can be solid, gaseous, and liquid, in a metastable parent phase. Most nucleation occurs heterogeneously unless the metastable parent phase from which the nuclei form is perfectly homogeneous and isolated from any catalyzing medium. This thesis project deals with heterogeneous nucleation on convex and concave spherical surfaces. In brief, the main achievements of the project are • An innovative analytical thermodynamic approach has been invented, which enabled rigorous thermodynamic formation of the energy barrier to nucleation, the critical radius and the shape factor, for nucleation on both convex and on concave surfaces. The rigorous thermodynamic analyses conducted have revealed a number of features for heterogeneous nucleation on convex and concave spherical surfaces as opposed to heterogeneous nucleation on a flat substrate surface. These are described in detail in Chapters 2 and 4. • Nucleation is the easiest on a concave spherical surface while it is the most difficult on a convex spherical surface assuming the contact angle and the critical embryo radius are the same. Nucleation on a flat substrate surface falls in between. This is determined by their shape factors. • The ratio R = 20r*, where R is the radius of the spherical substrate and r* is the critical embryo radius, (always define the symbols when they are first used. No one knows what they mean. R could be gas constant) can be regarded as a sufficiently accurate boundary that distinguishes between spherical and flat substrates for heterogeneous nucleation. • The investigation of the growth of crystal nuclei on a convex spherical surface has revealed that no growth barrier exists to the growth of a nucleus on a convex spherical substrate surface regardless of R < r* or R > r*. All nuclei formed on a convex spherical substrate surface are thus transformation nuclei. Turnbull’s transformation nucleus model or the recently developed free growth model does not apply to the growth of a spherical-cap nucleus on a convex spherical surface. • For heterogeneous nucleation in undercooled liquid metals, the cap thickness varies in a very narrow range by just a few angstroms and is typically about a few atomic layers thick according to Turnbull’s nucleation rate equation. The variations in the cap thickness are generally limited to less than 1Å when the contact angle Ɵ is varied in the range from 0° to 45°. It is anticipated that these findings will help to better understand the classical models for heterogeneous nucleation and provide new insights into the control of heterogeneous nucleation where convex or concave spherical substrate surfaces are involved.