\In this talk, I will describe my recent work with R. Mendes (Notre Dame) which begins a systematic study of closed manifolds with “strongly positive curvature”. This is an intermediate curvature condition between positive sectional curvature (sec>0) and positive curvature operator, and essentially dates back to the work of Thorpe in the early 70’s. We also study which of the known examples of manifolds with sec>0 also have this stronger condition, in particular among homogeneous spaces and biquotients.