A Fano manifold is a special type of complex manifold that has positive curvature properties. These manifolds are important in algebraic geometry and are characterized by having ample anticanonical bundles. This means that they can be embedded into projective space, making them useful for various geometric constructions. Fano manifolds often serve as examples in the study of Kähler geometry and mirror symmetry. They play a crucial role in the classification of algebraic varieties and have applications in theoretical physics, particularly in string theory, where they help in understanding compactifications and dualities.