Our understanding of the idea of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative differential geometry started when a Gauss-Bonnet-type theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansion suggest a general way of defining local curvature invariants for noncommutative Riemannian-type spaces where the metric structure is encoded by a Dirac-type operator. To carry out explicit computations, one needs a quite intriguing new set of ideas that are completely absent in the classical setting. In this talk, I shall try to highlight what has happened in the subject by giving an account of the most recent developments. For more details, I refer to https://arxiv.org/abs/1901.07438 and references therein.