[en] We study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks. We first determine the exact degree distribution of finite networks by generating function approaches. The resulting distributions exhibit an unusual finite-size scaling behaviour and they are also sensitive to the initial conditions. We argue that fluctuations in the number of nodes of degree k become Gaussian for fixed degree as the size of the network diverges. We also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution