In this tutorial we summarize the physics and mathematics behind refractive electromagnetic wave bending and delay. Refractive bending and delay through the Earth's atmosphere at both radio/millimetric and optical/IR wavelengths are discussed, but with most emphasis on the former, and with Atacama Large Millimeter Array (ALMA) applications in mind. As modern astronomical measurements often require sub-arcsecond position accuracy, care is required when selecting refractive bending and delay

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... ithms. For the spherically-uniform model atmospheres generally used for all refractive bending and delay algorithms, positional accuracies ≲ 1^" are achievable when observing at zenith angles ≲ 75^∘. A number of computationally economical approximate methods for atmospheric refractive bending and delay calculation are presented, appropriate for astronomical observations under these conditions. For observations under more realistic atmospheric conditions, for zenith angles ≳ 75^∘, or when higher positional accuracy is required, more rigorous refractive bending and delay algorithms must be employed. For accurate calculation of the refractive bending, we recommend the Auer & Standish (2000) method, using numerical integration to ray-trace through a two-layer model atmosphere, with an atmospheric model determination of the atmospheric refractivity. For the delay calculation we recommend numerical integration through a model atmosphere.