We discuss the unusual properties of networks that grow by either: (a) redirection or (b) node duplication. The former leads to unusual network properties when the redirection probability equals 1. For example, the number of nodes of degree greater than 1 scales slower than linearly in the total number of nodes N. In the latter case, a new node attaches to a randomly selected target node and also to each of its neighbors with probability p. The resulting network is sparse for p < 1/2 and dense (average degree increasing with number of nodes N) for p ≥ 1/2. The dense regime is especially rich. Individual network realizations are not self-averaging. There is also an infinite sequence of structural anomalies at p = 2/3, 3/4, 4/5, etc., where the N dependences of the number of triangles (3-cliques), 4-cliques, undergo phase transitions. When linking to second neighbors of the target can occur, the probability that the resulting graph is complete as N → ∞. This is collaborative work with U. Bhat, P.L. Krapivsky, and R. Lambiotte.