Various mathematical results on the so called Kirchhoff equations are reviewed and several techniques employed are confronted. After a short exposition of the original work of Kirchhoff, the three main kinds of existence results are discussed as follows: i) the local existence in Sobolev spaces, ii) the global existence for analytic initial data, and iii) the global existence for small and compactly supported data. This note is very convenient to study Kirchhoff equations for the first time.