Complex surfaces are mathematical objects studied in the field of complex geometry. They are defined as two-dimensional manifolds that can be described using complex numbers. These surfaces can exhibit intricate shapes and structures, often arising from the study of algebraic varieties and complex analytic spaces. One important aspect of complex surfaces is their classification, which involves understanding their topological and geometric properties. Notable examples include the K3 surfaces and Calabi-Yau manifolds, both of which play significant roles in string theory and algebraic geometry. The study of complex surfaces helps mathematicians explore deeper connections between geometry, topology, and algebra.