Strongly positive curvature is an intermediate condition between positive-definiteness of the curvature operator and positive sectional curvature, defined in terms of modifying the curvature operator with a 4-form to make it positive-definite. In this talk, I will discuss some background and two recent results on the subject: the classification of homogeneous spaces with strongly positive curvature, and the verification that all manifolds known to admit metrics with nonnegative sectional curvature also admit metrics with strongly nonnegative curvature. I will also report on work in progress using the Bochner technique in this context to find new topological obstructions. This is based on joint work with R. Mendes (WWU Muenster).