When does a manifold admit a metric with positive sectional curvature? This is one of the most fundamental and difficult problems in differential geometry. One attempt at understanding this problem is to begin with a non-negatively curved manifold and examine how large is the set of points with positive curvature. More precisely, given a manifold, does it admit a metric for which there is an open set of points with positive curvature (quasi-positive curvature), or a metric with an open dense set of such points (almost positive curvature)? We present the known examples with quasi and almost positive curvature and present some new ones.