I will begin with the question: Classify the left-invariant metrics with nonnegative sectional curvature on a given compact Lie group. I will then discuss applications of the natural power-series approach to this problem, including a partial classification of cohomogeneity-one manifolds with nonnegative curvature and at least one totally geodesic principle orbit and new examples of homogeneous metrics with nonnegative curvature. This work is joint with Lorenz Schwachhoefer.