This volume is an amplification of the Colloquium Lectures delivered under the title ""The Determination of the Tritangent Planes of the Space Sextic of Genus Four"". In order to present clearly the state of that problem, a comparison with the better known cases of genus two and genus three is desirable. Preliminary chapters on algebraic geometry and theta functions are incorporated in order to facilitate reading by recalling fundamental ideas of these two subjects in such fashion as will be most helpful in later applications.