  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 algorithms. For the
spherically-uniform model atmospheres generally used for all refractive bending
and delay algorithms, positional accuracies $\lesssim 1^{\prime\prime}$ are
achievable when observing at zenith angles $\lesssim 75^\circ$. 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 $\gtrsim 75^\circ$, 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.
