We study representations of Lie groups in Banach spaces, particularly distribution representations (or smeared representations) and differentiable representations. These objects, which we define, are shown to be natural and useful generalizations of the classical unitary representations and the point strong operator continuous representations. A distribution (smeared) representation is a homomorphism of the convolution algebra of compactly supported P-functions on a Lie group, G say, into the algebra of bounded operators on a Banach space, satisfying certain natural additional conditions. It is shown that if I’ is a differentiable representation and d is the Laplace operator on G, then V can be integrated to a smeared representation of G under certain conditions on the resolvent set of the unbounded operator U(d). A partial converse is also shown. Finally we give conditions under which the operators V(y) are of trace class where V is a smeared representation and q~ is a compactly supported Cm-function on G.