Manifolds that curve like a sphere are called positively curved, as opposed to those that curve like a saddle, which are called negatively curved. Manifolds with almost positive curvature are positively curved at almost every point in a probabilistic sense, and they are highly sought. The most recent discovery of a positively curved manifold was 2008, and no infinite family of almost positively curved manifolds has been discovered since 2002 until now. We construct infinitely many new examples of manifolds with positive curvature almost everywhere.