Classical and Quantum $V$-algebras
  The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all $H$-reduced $\hat{\cal G}_{r}$-current algebras belong to one of the following basic algebraic structures: local quadratic $W$-algebras, rational $U$-algebras, nonlocal $V$-algebras, nonlocal quadratic $WV$-algebras and rational nonlocal $UV$-algebras. The main new features of the quantum $V$-algebras and their heighest weight representations are demonstrated on the example of the quantum $V_{3}^{(1,1)}$-algebra. 
J.F.Gomes, F.E. Mendon\c{c}a da Silveira, G.M. Sotkov and A.H.   Zimerman