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Möbius strips are prototypical examples of ribbon-like structures. Inspecting their shapes and features provides useful insights into the rich mechanics of elastic ribbons. Despite their ubiquity and ease of construction, quantitative experimental measurements of the three-dimensional shapes of Möbius strips are surprisingly non-existent in the literature. We propose two novel stereo vision-based techniques to this end—a marker-based technique that determines a Lagrangian description for the construction of a Möbius strip, and a structured light illumination technique that furnishes an Eulerian description of its shape. Our measurements enable a critical evaluation of the predictive capabilities of mechanical theories proposed to model Möbius strips. We experimentally validate, seemingly for the first time, the developable strip and the Cosserat plate theories for predicting shapes of Möbius strips. Equally significantly, we confirm unambiguous deficiencies in modelling Möbius strips as Kirchhoff rods with slender cross-sections. The experimental techniques proposed and the Cosserat plate model promise to be useful tools for investigating a general class of problems in ribbon mechanics.