1. Functions and Measures in Linear Spaces.- 1. Spaces, Mappings, Differential Operations.- 2. Measures and Integrals.- 3. Measure Differentiation.- 2. Functions and Measures on Smooth Manifolds.- 1. Smooth Manifolds and Vector Bundles.- 2. Bundle Section, Connections, Differential Operations.- 3. Hilbert and Hilbert-Schmidt Bundles.- 4. Measures on Smooth Manifolds.- 3. Stochastic Equations in Banach Spaces.- 1. Basic Notions.- 2. Stochastic Integrals for Vector and Operator Functions.- 3. Stochastic Equations.- 4. Multiplicative Functionals of Stochastic Processes.- 5. Stochastic Flow.- 4. Stochastic Equations on Smooth Manifolds.- 1. Stochastic Differentials.- 2. Stochastic Differential Equations on Manifolds.- 3. Stochastic Equations in Vector Bundles.- 5. Kolmogorov Equations.- 1. Backward Kolmogorov Equations.- 2. Quasilinear Parabolic Equations.- 3. Forward Kolmogorov Equations.- 6. Diffusion Processes on Lie Groups and Principal Fibre Bundles.- 1. Lie Groups and Smooth Bundles.- 2. Invariant Stochastic Equations.- 3. Stochastic Equations on Manifolds and their Solution Distribution Properties.- Historical Comments.- References.