The distinction between axiomatic demonstration and analytic demonstration provides a basis for dealing with the question of mathematical explanations. One may speak of mathematical explanations in two different senses: mathematical explanations of mathematical facts and mathematical explanations of empirical facts. This chapter mainly deals with mathematical explanations of mathematical facts. It argues that there is an objective distinction between explanatory and non-explanatory demonstrations, and distinguishes between two different approaches to explanatory demonstrations: the static approach, based on axiomatic demonstration, and the dynamic approach, based on analytic demonstration. The chapter maintains that explanatory demonstrations in the static approach are intended to convince the audience that a proposition should be accepted, while explanatory demonstrations in the dynamic approach are intended to reveal how the demonstration was discovered. The chapter also deals with mathematical explanations of empirical facts, maintaining that it is unjustified to claim that there are such explanations.