Tremendous progress in computing perturbative scattering amplitudes in N=4 supersymmetric gauge theory has been made over the past few years. Importantly the planar amplitudes appear to display a dual conformal invariance next to the usual conformal symmetry. Altogether the symmetry enlarges to a Yangian algebra known from the context of integrable models. This infinite-dimensional symmetry might have the power to completely fix the S-matrix by algebraic means. In this talk we review the above developments. We then discuss conformal symmetry for tree and loop scattering amplitudes. It turns out that the free conformal symmetry generators are anomalous which calls for certain deformations to make the symmetries exact. These relate amplitude with different numbers of legs, and thus they contribute substantially to a complete algebraic determination.