  A superposition rule is a particular type of map that enables one to express
the general solution of certain systems of first-order ordinary differential
equations, the so-called Lie systems, out of generic families of particular
solutions and a set of constants. The first aim of this work is to propose
several generalisations of this notion to second-order differential equations.
Next, several results on the existence of such generalisations are given and
relations with the theories of Lie systems and quasi-Lie schemes are found.
Finally, our methods are used to study second-order Riccati equations and other
second-order differential equations of mathematical and physical interest.
