Banking disintermediation induces a greater role for shadow banking firms, among which investment funds. These funds attract a large part of savings that were previously channelled by banks. This evolution justifies scrutinizing the behavior of those agents and identifying the risks that they bear. We review theoretical and empirical literature that shows evidence for the risks caused by delegated monitoring, herding, runs and first-mover advantage. We propose a test of run risk for several French mutual funds (2005-2015), and show that funds invested in bonds exhibit a first mover advantage. We do not find such relation for funds invested in stocks.