We give some results on the universal fibration X \to UE \to Baut_1(X) for fibrations with fibre X. When X is an elliptic space with evenly graded rational cohomology satisfying Halperin's conjecture, we prove the universal fibration has rational sectional category equal to two. When X has finite-dimensional rational homotopy, we give a dg Lie model for the evaluation fibration for Baut_1(X). We apply the latter result to the problem of realizing spaces as classifying spaces and that of iterating the classifying space functor. This is joint work with Greg Lupton.