This semi-tutorial paper introduces the partial-inverse problem for polynomials and develops its application to decoding Reed-Solomon codes and some related codes. The most natural algorithm to solve the partial-inverse problem is very similar to, but more general than, the Berlekamp-Massey algorithm. Two additional algorithms are obtained as easy variations of the basic algorithm: the first variation is entirely new, while the second variation may be viewed as a version of the Euclidean algorithm. Decoding Reed-Solomon codes (and some related codes) can be reduced to the partial-inverse problem, both via the standard key equation and, more naturally, via an alternative key equation with a new converse. Shortened and singly-extended Reed-Solomon codes are automatically included. Using the properties of the partial-inverse problem, two further key equations with attractive properties are obtained. The paper also points out a variety of options for interpolation.