This chapter is concerned with some of the many generalizations of classical Lie symmetries, i.e., conditional symmetries and equivalence transformations. Conditional symmetries generalize the nonclassical symmetries introduced in 1969 by Bluman and Cole investigating the symmetries of the solutions of the Fourier equation constrained to be also solutions of the invariant surface condition. The procedure for obtaining nonclassical symmetries is outlined, and a short account of the Clarkson and Kruskal direct method is given. The more general concept of conditional symmetry, or weak symmetry, is also discussed. Starting with Burgers equation, and repeatedly searching for nonclassical symmetries, a hierarchy of systems of coupled Burgers-like equations is derived. Moreover, it is shown how some classical results known as substitution principles in gas dynamics and magneto gas dynamics can be interpreted as weak symmetries. Conditional symmetries are widely used for determining exact solutions of equations coming from applications. On the contrary, equivalence transformations, extensively studied by Ovsiannikov (but also considered by Lie himself), constitute a nice framework for investigating the invariance of classes of differential equations containing unspecified functions.