I will discuss old and new results on the following questions: Given a connected Lie group G and a manifold M, does G have an effective action on M? If so, how smooth can it be? What can be said about fixed points? Sample theorems: * The group of 2 x 2 upper triangular real matrices with positive diagonals has an effective real analytic action, with finite fixed point set, on every compact surface. * Let G be a group such that in the adjoint representation of its Lie algebra some element has a nonreal eigenvalue. If G acts effectively and analytically on a compact surface, the Euler characteristic is an upper bound for the number of fixed points. Thus the surface has nonnegative Euler characteristic. But there are such groups with effective continuous actions on all surfaces. * The universal covering group of SL(2, R) acts effectively with discrete fixed point set on every manifold. * The affine group on the line acts effectively on S^2 without fixed point. But such an action cannot be analytic. * If the Lie algebra of G has a basis X, U, V, Z with Z central and [X,U] = V, [X,V] = -U, [U, V] = Z, then G does not have an effective action on any surface.