This paper explores fractional Sobolev spaces \( W^{s,p} \) and their definitions, properties, and applications. The authors analyze the relationships between different definitions of these spaces and their role in trace theory. They prove continuous and compact embeddings, investigate extension domains, and present regularity results. The paper also includes counterexamples in non-Lipschitz domains. The authors aim to provide a self-contained guide for students and young researchers, covering topics such as the Gagliardo norm, fractional Laplacian, and Sobolev inequalities. They avoid interpolation techniques and Besov space theory, offering original proofs. The paper is structured into sections that cover the definition of fractional Sobolev spaces, their properties, the Hilbert case, asymptotic behavior of the constant factor in the fractional Laplacian, extension problems, Sobolev inequalities, compact embeddings, and counterexamples in non-Lipschitz domains.This paper explores fractional Sobolev spaces \( W^{s,p} \) and their definitions, properties, and applications. The authors analyze the relationships between different definitions of these spaces and their role in trace theory. They prove continuous and compact embeddings, investigate extension domains, and present regularity results. The paper also includes counterexamples in non-Lipschitz domains. The authors aim to provide a self-contained guide for students and young researchers, covering topics such as the Gagliardo norm, fractional Laplacian, and Sobolev inequalities. They avoid interpolation techniques and Besov space theory, offering original proofs. The paper is structured into sections that cover the definition of fractional Sobolev spaces, their properties, the Hilbert case, asymptotic behavior of the constant factor in the fractional Laplacian, extension problems, Sobolev inequalities, compact embeddings, and counterexamples in non-Lipschitz domains.