We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-Émery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the Möbius ladders. We also highlight an online tool for calculating the curvature of graphs under several variants of the curvature notions that we use in the classification. As a consequence of the classification result we show that non-negatively curved cubic expanders do not exist.arXiv:1712.03033v2 [math.CO]

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