Introduction Manifold as a surface in Rn. Tangent space, vector fields, interval, area and volume on an embedded surface Non-relativistic and relativistic particles, strings and D-brans Mappings of manifolds Examples of mappings (embeddings, projections, diffeomorphisms) Differential of mapping (behaviour of tangent vectors under mappings) Vector fields as infinitesimal diffeomorphisms Commutator of vector fields Lie algebra of vector fields, conformal invariance of a string Local coordinates Mapping in local coordinate notation Global and local properties Concept of covariance Transformation of vector fields under replacement of coordinates Invariance of the commutator of vector fields Independence of the physical content of theory from a reference system (Einstein principle of equivalence) Tensor fields Examples of tensor fields (metric tensor, Poisson and symplectic structures, complex structure) Behaviour of tensor fields under mappings Lie derivative concept, invariant vector fields, symmetry, Killing vectors Covariance of the Poisson and symplecticity conditions Symplectic manifolds Darboux's theorem Hamiltonoan systems, Hamilton equations, Hamiltonian fluxes, canonical transformations Differential forms and operations over it Exterior algebra of manifold de Rham differential, closed and exact forms Poincare lemma, Hodge conjugation Maxwell equations in differential form notation Self-duality equation Superfields, supersymmetry, superspace General concept of manifold Atlas, maps Global and local properties de Rahm complex, cohomologies Topological interpretation of a dot charge and Aharonov-Bohm effect Lie algebras, Lie algebra of vector fields Groups of transformations and its Lie algebras Lie groups (abstract definition) Lie groups as groups of transformations Left- and right-invariant vector fields on group Symmetries of physical systems (rotations, Lorentz and Poincare group, relativistic invariance, symmetry group of AdS space, conformal symmetry, internal symmetries) Vector bundles Bundle cross-sections Structural group of vector bundle Principal bundle associated with vector one Tangent bundle and cotangent bundle, bundle of local reference points Operations over bundles Tensor fields as cross-sections of corresponding bundles Trivial and non-trivial bundles Parallelizable manifolds Connection in vector (principal) bundle Connection form and curvature Maurer-Cartan equation, parallel transport Yang-Mills fields Gauge transformations Physical applications (transition from a global internal symmetry to a gauge one) Invariants of gauge transformations, Yang-Mills action Integration on manifolds, volume form Stoks theorem, densities and semidensities Volume forms on Riemannian and symplectic manifolds, Liuville form Actions for dynamic systems on general manifolds Chern-Simon action and other topological models Riemannian geometry, metric connection, curvature GRT (action, Einstein equations, general coordinate invariance) Tetradic Einstein-Cartan formalism, supergravitaty Almost complex and complex structures Niewenhuizen torsion tensor Holomorphic and antiholomorphic functions Tensor fields on complex manifolds Dolboux differentials Bi-complex Kahlerian manifolds Riemannian surfaces Conformal transformations, conformal invariance Kahlerian S -model Elements of supergeometry Supermanifolds and super Lie algebras Antibracket and D -operator, Batalin-Vylkovsky algebra Polyvectors and differential forms, Witten realization of the Batalin-Vylkovsky algebra