This paper provides an overview of fractional Sobolev spaces $ W^{s,p} $, analyzing their definitions, properties, and applications. The authors define these spaces using the Gagliardo norm and investigate their role in trace theory. They prove continuous and compact embeddings, discussing extension domains and regularity results. The paper also presents counterexamples in non-Lipschitz domains and explores the relationship between fractional Sobolev spaces and the fractional Laplacian operator. Key results include the equivalence of the Gagliardo norm with the Fourier transform-based definition, the asymptotic behavior of the constant $ C(n,s) $, and the embedding of fractional Sobolev spaces into Hilbert spaces. The authors emphasize the importance of these spaces in various applications, such as thin obstacle problems, optimization, finance, and materials science. The paper is written in a self-contained, accessible style, avoiding advanced interpolation techniques and focusing on clarity and simplicity. It serves as a comprehensive guide for students and researchers interested in fractional calculus and its applications.This paper provides an overview of fractional Sobolev spaces $ W^{s,p} $, analyzing their definitions, properties, and applications. The authors define these spaces using the Gagliardo norm and investigate their role in trace theory. They prove continuous and compact embeddings, discussing extension domains and regularity results. The paper also presents counterexamples in non-Lipschitz domains and explores the relationship between fractional Sobolev spaces and the fractional Laplacian operator. Key results include the equivalence of the Gagliardo norm with the Fourier transform-based definition, the asymptotic behavior of the constant $ C(n,s) $, and the embedding of fractional Sobolev spaces into Hilbert spaces. The authors emphasize the importance of these spaces in various applications, such as thin obstacle problems, optimization, finance, and materials science. The paper is written in a self-contained, accessible style, avoiding advanced interpolation techniques and focusing on clarity and simplicity. It serves as a comprehensive guide for students and researchers interested in fractional calculus and its applications.