  We discuss general properties of $A_\infty$-algebras and their applications
to the theory of open strings. The properties of cyclicity for
$A_\infty$-algebras are examined in detail. We prove the decomposition theorem,
which is a stronger version of the minimal model theorem, for
$A_\infty$-algebras and cyclic $A_\infty$-algebras and discuss various
consequences of it. In particular it is applied to classical open string field
theories and it is shown that all classical open string field theories on a
fixed conformal background are cyclic $A_\infty$-isomorphic to each other. The
same results hold for classical closed string field theories, whose algebraic
structure is governed by cyclic $L_\infty$-algebras.
