  We develop geometric approach to A-infinity algebras and A-infinity
categories based on the notion of formal scheme in the category of graded
vector spaces. Geometric approach clarifies several questions, e.g. the notion
of homological unit or A-infinity structure on A-infinity functors. We discuss
Hochschild complexes of A-infinity algebras from geometric point of view. The
paper contains homological versions of the notions of properness and smoothness
of projective varieties as well as the non-commutative version of Hodge-to-de
Rham degeneration conjecture. We also discuss a generalization of Deligne's
conjecture which includes both Hochschild chains and cochains. We conclude the
paper with the description of an action of the PROP of singular chains of the
topological PROP of 2-dimensional surfaces on the Hochschild chain complex of
an A-infinity algebra with the scalar product. This action is essentially
equivalent to the structure of 2-dimensional Topological Field Theory
associated with a Calabi-Yau category.
