let Γ _n denote the group of oriented homotopy n-spheres, n>4. There is a decreasing filtration of Γ _n due to Gromoll: Γ _n = Γ _{{n, 2}} ⊃ Γ _{{n, 3}} ⊃ ... ⊃ Γ _{{n, n-3}} = 0 The determination of the Gromoll filtration is a difficult open problem (it seems to be unknown for any n > 6, n ≠ 12): it has many interesting applications in geometric topology. In this talk I will survey approaches to finding elements deep in the Gromoll filtration: plumbing, compostion and pseudo-isotopy theory and I will report on new computations for certain Gromoll groups. As time permits I will discuss applications and open problems.