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The book is devoted to study the geometry of smooth surfaces and the geometry of intersecting hypersurfaces. The study comprises of five chapters. The first chapter is introductory in nature. The second chapter is devoted to the study of smoothness of surfaces. It consists of two sections. In the first section we study the second order smoothness measure and in the second section we study the third order smoothness measure. The third chapter is devoted to study of geometry of intersecting curves of surfaces in Euclidean-3 spaces. The analysis of chapter 3 leads us to study geometry of intersecting curves in higher dimension Euclidean spaces. In chapter 4 we study the geometry of transversal intersection of curves of hypersurfaces in Euclidean-4 spaces. The last chapter of the book is devoted to study the geometry of non-transversal intersection of hypersurfaces in R4. In the first section we study the geometry of non-transversal almost tangential intersection curves of parametric-parametric-parametric hypersurfaces. In the second section we study the geometry of non-transversal intersection curves of parametric-parametric-parametric hypersurfaces.