Character varieties parametrize representations of the fundamental group of a variety into complex Lie groups. Of special interest for 3-manifold topology is the case of the complement of a knot in the 3-sphere. We shall study the case of the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C), giving explicit equations. This character variety has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from Sym^2:SL(2,C)->SL(3,C). The other two components are induced by
exceptional Dehn fillings of the figure eight knot (joint work with Michael Heusener and Joan Porti, arxiv:1505.04451).