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Alfred Schnittke was a Russian composer who is regarded as one of the leading avant-garde composers of our time; he recently passed away in 1998. His prolific works were eclectic in a wide variety of genres and styles. He was particularly known for polystylicism, a coined word to describe a combination of different styles. Among such works are his piano sonatas, which deserve study and performance since they also represent an important contribution to the continuing genre in the twentieth-century. This document gives brief analyses of his first two piano sonatas. After Schnittke's biography and discussions of his major works, the text of investigation contains the introduction, either cyclic themes or thematic similarities, analysis of each movement, and comparison of the sonatas. The analyses consist of discussions of form, melody, harmony, rhythm, texture, range, and special effects. The first sonata is constructed on Vladimir Feltsman's musical monogram and cyclic themes. The second sonata is based on themes that are related to one another. Schnittke loosely applied the term, sonata, as did so many other composers of this era. The sonatas lack the traditional concept of the formal structure associated with themes in certain keys. They both exhibit a wide range of moods and characters including climactic conclusions. This document also contains a bibliography and an appendix of a list of his works and writings.