Consider the following contrasting results: Theorem A. [B. E. Johnson, 1994 + R. J. Plymen, unpublished] Fourier algebras of compact, connected, non-abelian Lie groups always support non-zero derivations into suitable symmetric target modules. Theorem B. [N. Spronk, 2002] If we consider arbitrary locally compact groups, but now require derivations to be completely bounded and the symmetric targets to be completely contractive as modules, then there are no non-zero derivations. I will summarise some work from the last five years, by various authors, which shows that we can drop the word ``compact'' from Theorem A, thus resolving the Lie case of a conjecture by B. E. Forrest and V. Runde. Explicit calculations using the non-abelian Fourier transform and Plancherel theorem play a crucial rule. My own contributions here are joint work with M. Ghandehari. I will then present some work in progress, which shows inter alia that there are groups whose Fourier algebras are not 2-dimensionally weakly amenable in the sense of [Johnson, 1997]. Although the 2-cocycles constructed for this purpose are not completely bounded, their construction requires some operator-space machinery which appears to be new; Theorem B is an obstacle to the naive attempt. If time remains, I will sketch some arguments that suggest the Fourier algebra of the three-dimensional Lie group SU(2) is 3-dimensionally weakly amenable.