Laplace operator and the heat equation in $\mathbb{R}^n$ Function spaces in $\mathbb{R}^n$ Laplace operator on a Riemannian manifold Laplace operator and heat equation in $L^{2}(M)$ Weak maximum principle and related topics Regularity theory in $\mathbb{R}^n$ The heat kernel on a manifold Positive solutions Heat kernel as a fundamental solution Spectral properties Distance function and completeness Gaussian estimates in the integrated form Green function and Green operator Ultracontractive estimates and eigenvalues Pointwise Gaussian estimates I Pointwise Gaussian estimates II Reference material Bibliography Some notation Index