Summary: Some recent work in Fréchet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case is extended to infinite dimensional Banach manifolds and Fréchet manifolds that could be represented as projective limits of Banach manifolds. This leads to further results concerning the characterization of second tangent bundles and differential equations in the more general Fréchet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Fréchet spaces.