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 ^{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 ; Appendix A. Reference material