Young measures are tools that allow one to describe limits of nonlinear quantities depending on weakly converging sequences,a recurring problem in the Calculus of Variations and nonlinear PDE.It is important to understand which are the Young measures that can be generated by sequences of gradients of Sobolev maps,a question answered by Kinderlehrer & Pedregal in 91 and 94.However,determinant constraints on the gradients which are important for elasticity are non-convex and cannot be handled.In this talk,I will present a characterization result in the spirit of Kinderlehrer & Pedregal for Young measures generated by gradients satisfying various determinant constraints,including orientation-preserving maps.The techniques involved are novel and use a variant of convex integration,providing generating sequences in Sobolev spaces of exponent necessarily less than the dimension where we have the required flexibility to "correct" the value of the Jacobian.Interesting applications will also be presented.This work is joined with F.Rindler and E.Wiedemann.