A dessin d'enfant can be thought of as a conjugacy class of homomorphisms m:Pi --> S_n, where Pi is the fundamental group of the thrice-punctured line and S_n is the symmetric group on n letters. Belyi's theorem tells us that Gal(Qbar/Q) injects into Out(Pi); thus, Gal(Qbar/Q) acts on the set of all dessins d'enfants. A basic (the basic?) problem in the theory of dessins d'enfants is the analysis of the orbits of this action. In particular: when are two dessins d'enfant Galois-conjugate? One way to distinguish non-conjugate dessins is by means of invariants; for instance, the _image_ of m is obviously unchanged by Galois conjugation. We will discuss some known invariants for the action of Gal(Qbar/Q), which we show are also invariants for the a priori larger group GT^hat. We will also discuss a new Galois invariant, related to the spherical braid group, which is not obviously a GT^hat invariant.