The chapter introduces the basic concepts of enriched category theory, focusing on monoidal categories and their applications. A monoidal category \(\mathcal{V}\) consists of a category \(\mathcal{V}_0\), a functor \(\otimes : \mathcal{V}_0 \times \mathcal{V}_0 \longrightarrow \mathcal{V}_0\), an object \(I\) of \(\mathcal{V}_0\), and natural isomorphisms \(a, l, r\) satisfying coherence axioms. Monoidal categories include cartesian monoidal categories, such as \(\mathbf{Set}\), \(\mathbf{Cat}\), and \(\mathbf{Top}\), as well as non-cartesian examples like \(\mathbf{Ab}\), \(\mathbf{R-Mod}\), and \(\mathbf{DG-R-Mod}\). The 2-category \(\mathcal{V}\)-CAT is defined for a given monoidal \(\mathcal{V}\), consisting of \(\mathcal{V}\)-categories, \(\mathcal{V}\)-functors, and \(\mathcal{V}\)-natural transformations. The forgetful 2-functor \((\cdot)_0 : \mathcal{V}\)-CAT \longrightarrow\) CAT maps a \(\mathcal{V}\)-category to its underlying ordinary category. The chapter also discusses symmetric monoidal categories, where a symmetry \(c\) is a natural isomorphism satisfying coherence axioms, and the tensor product and duality in \(\mathcal{V}\)-CAT. Closed and biclosed monoidal categories are introduced, where each functor \(- \otimes Y\) has a right adjoint \([Y, -]\). The chapter highlights the importance of these structures in enriched category theory and their applications in various mathematical fields.The chapter introduces the basic concepts of enriched category theory, focusing on monoidal categories and their applications. A monoidal category \(\mathcal{V}\) consists of a category \(\mathcal{V}_0\), a functor \(\otimes : \mathcal{V}_0 \times \mathcal{V}_0 \longrightarrow \mathcal{V}_0\), an object \(I\) of \(\mathcal{V}_0\), and natural isomorphisms \(a, l, r\) satisfying coherence axioms. Monoidal categories include cartesian monoidal categories, such as \(\mathbf{Set}\), \(\mathbf{Cat}\), and \(\mathbf{Top}\), as well as non-cartesian examples like \(\mathbf{Ab}\), \(\mathbf{R-Mod}\), and \(\mathbf{DG-R-Mod}\). The 2-category \(\mathcal{V}\)-CAT is defined for a given monoidal \(\mathcal{V}\), consisting of \(\mathcal{V}\)-categories, \(\mathcal{V}\)-functors, and \(\mathcal{V}\)-natural transformations. The forgetful 2-functor \((\cdot)_0 : \mathcal{V}\)-CAT \longrightarrow\) CAT maps a \(\mathcal{V}\)-category to its underlying ordinary category. The chapter also discusses symmetric monoidal categories, where a symmetry \(c\) is a natural isomorphism satisfying coherence axioms, and the tensor product and duality in \(\mathcal{V}\)-CAT. Closed and biclosed monoidal categories are introduced, where each functor \(- \otimes Y\) has a right adjoint \([Y, -]\). The chapter highlights the importance of these structures in enriched category theory and their applications in various mathematical fields.