This book contains a written version of lectures on Kähler geometry prepared for graduate students of mathematics and theoretical physics from a perspective of Riemannian geometry. The first part explains basic notions from differential geometry (manifolds, tensor fields, integration of differential forms, principal and vector bundles, connections, Riemannian manifolds, parallel transport and a concept of holonomy). The second part is devoted to complex and Hermitean geometry (complex manifolds, complex and holomorphic vector bundles, Hermitean vector bundles, the Chern connection, Kähler metrics, the Kählerian curvature tensor, the Laplace operator on a Kähler manifold, Hodge theory and Dolbeault theory).
Special and more advanced topics are discussed in the last part of the book, including a short description of the first Chern classes of vector bundles, Ricci-flat Kähler manifolds, the Calabi-Yau theorem, the Aubin-Yau theorem on Kähler-Einstein metrics, vanishing theorems on Kähler manifolds, the Hirzebruch-Riemann-Roch formula, hyperkähler manifolds and Calabi-Yau manifolds). Recent intensive collaboration between mathematics and theoretical physics in areas connected with string theory has, as a consequence, created substantially stronger requirements on the mathematical background of young people aiming to work in this broad field. The book covers in a nice, systematic and understandable way basic tools of differential geometry. It can be useful both for student readers and for teachers preparing lectures on the subject.