Geometry, topology and broken symmetry often play a powerful role in determining the organization and properties of materials. A recent example is the discovery that the excitation spectra of materials -- be they electronic, optical, or mechanical -- may be topologically non-trivial. I will explore the use of `spinning tops' to explore this physics. In particular I will discuss an experimental and theoretical study of a simple kind of active meta-material – coupled gyroscopes – that naturally encodes non-trivial topology in its vibrational spectrum. These materials have topologically protected edge modes which we observe in experiment. Crucially, the geometry of the underlying lattice controls the presence of time reversal symmetry that is essential to the non-trivial topology of the spectrum. We exploit this to control the chirality of the edge modes by simply deforming the lattice. Moving beyond ordered lattices we show that amorphous gyroscopic networks are naturally topological. We construct them from arbitrary point sets -- including hyperuniform, jammed, quasi-crystalline and uniformly random -- and control their topology through simple, local decorations.