Non-skew-symmetric classical r-matrices and integrable Hamiltonian systems

In the present talk we will review a theory of classical non-skew-symmetric non-dynamical  r-matrices  with spectral parameters and their usage  the theory of integrable classical and quantum systems.   We will explain the relation of these r-matrices with the theory of  infinite-dimensional almost-graded  Lie algebras with  Kostant-Adler decomposition. We will  present several classes of examples of such the r-matrices, naturally lying out of the Belavin-Drinfeld classification. In particular, we will present classical r-matrices related to integrable multidimensional  tops  (Manakov tops). We will also outline a sub-class of the non-skew-symmetric classical r-matrices permitting to construct, except for the linear tensor brackets, also the quadratic tensor brackets that lead to Maillet and reflection equation algebras. We will  in details consider  Gaudin models with and without external magnetic field and their generalizations based on non-skew-symmetric  classical r-matrices. Applications of these models to the theory of isomonodromic deformations and to Knizhnik-Zamolodchikov-type  equations will be  discussed.