We introduce the notion of a derived Poisson structure on an associative
(not necessarily commutative) algebra A. This structure is characterized
by the property of being a "weakest" structure on A that induces natural
Poisson structures on the derived moduli spaces of finite-dimensional
representations of A. A derived Poisson structure gives rise to a graded
Lie algebra structure on cyclic homology; it can thus be viewed as a
higher homological extension of the notion of H_0-Poisson structure
introduced by W. Crawley-Boevey (2011). We will give new examples and
present recent results on Calabi-Yau algebras obtained in joint work with
X. Chen, F. Eshmatov and S. Yang. If time permits, we will explain the
relation between derived Poisson structures and the Chas-Sullivan
brackets arising in string topology.