The chemistry and catalysis of solid heteropoly compounds are described. Heteropoly compounds are heteropolyanions and their derivatives that maintain the polyanion structure (metal oxide clusters), and catalysts of which the compounds are the essential components are called heteropoly catalyst. Heteropoly compounds are called polyoxometalates as well. First, the structure and chemical properties of heteropoly compounds are summarized, emphasizing the acidic and redox properties. The polyanion structure can be preserved in the solid state, if appropriate cautions are paid, and the acid and redox properties can be controlled in a wide range by choosing the polyanion structure and constituent elements. These are the remarkable advantages of the heteropoly catalysts. Then the catalysis and catalyst design of the heteropoly compounds are described in detail. There are actually several industrial processes using heteropoly catalysts as solid acids and oxidation catalysts. It is important to recognize the presence of hierarchical structures (primary, secondary, and tertiary structures) and three major types of catalysis (surface-type and two bulk-type catalyses). Due to this, catalyst design becomes more realistic and unique properties like shape-selective catalysis develop. Moreover, the dynamic behavior of protons and organic molecules can be elucidated by spectroscopic techniques.