  We construct Adams operations on the cohomology theory Tmf of topological
modular forms; the first such stable operations on this cohomology theory.
These Adams operations are then calculated on the Tmf-cohomology of spheres
using a combination of descent spectral sequences and Anderson duality.
Applications of these operations are then given, including constructions of
connective height 2 analogues of Adams summands and image of J spectra.
