This paper contains two interrelated developments. First are proposed new generalized Verma modules. They are called κ-Verma modules, κ ϵ N , and coincide with the usual Verma modules for κ = 1. As a vector space a k-Verma module is isomorphic to the symmetric tensor product of κ copies of the universal enveloping algebra U( G −), where G − is the subalgebra of lowering generators in the standard triangular decomposition of simple Lie algebra G = G + ⊛ H ⊛ G − . The second development is the proposal of a procedure for the construction of multilinear intertwining differential operators for semisimple Lie groups G. This procedure uses κ-Verma modules and coincides for κ = 1 with a procedure for the construction of linear intertwining differential operators. For all κ a central role is played by the singular vectors of the k-Verma modules. Explicit formulae for series of such singular vectors are given. Using these are given explicitly many new examples of multilinear intertwining differential operators. In particular, for G = SL(2, R ) are given explicitly all bilinear intertwining differential operators. Using the latter, as an application are constructed 1 2n - differentials for all n ϵ 2 N , the ordinary Schwarzian being the case n = 4.