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The catenary is one of the most common curves; it is, in fact, the shape assumed by a homogeneous and inextensible chain, fixed to the extremities, which is subject only to its own weight. The catenary has always fascinated not only mathematicians but also architects and engineers, who have often used it in their works due to its remarkable properties. In this note, the catenary is introduced by determining its equation and considering its history from Galileo Galilei to the present day, pointing out a historical mistake. Then, some of its applications to architecture and engineering are shown and the catenary is compared to the parabola, a curve that at first sight can look similar to catenary but must not be confused with it. Finally, some variants of the catenary are considered: the weighted catenary and the catenary of equal resistance, highlighting their properties and their practical applications.