  We first review various known algebraic structures on the Hochschild
(co)homology of a differential graded algebras under weak Poincar{\'e} duality
hypothesis, such as Calabi-Yau algebras, derived Poincar{\'e} duality algebras
and closed Frobenius algebras. This includes a BV-algebra structure on
$HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of
the natural Gerstenhaber structure on $HH^*(A,A)$. As an example, after proving
that the chain complex of the Moore loop space of a manifold $M$ is a
CY-algebra and using Burghelea-Fiedorowicz-Goodwillie theorem we obtain a
BV-structure on the homology of the free space. In Sections 6 we prove that
these BV/coBVstructures can be indeed defined for the Hochschild homology of a
symmetric open Frobenius DG-algebras. In particular we prove that the
Hochschild homology and cohomology of a symmetric open Frobenius algebra is a
BV and coBV-algebra. In Section 7 we exhibit a BV structure on the shifted
relative Hochschild homology of a symmetric commutative Frobenius algebra. The
existence of a BV-structure on the relative Hochschild homology was expected in
the light of Chas-Sullivan and Goresky-Hingston results for free loop spaces.
In Section 8 we present an action of Sullivan diagrams on the Hochschild
(co)chain complex of a closed Frobenius DG-algebra. This recovers
Tradler-Zeinalian \cite{TZ} result for closed Froebenius algebras using the
isomorphism $C^*(A ,A) \simeq C^*(A,A^\vee)$.
