Flooding is among the simplest and most fundamental of all distributed network algorithms. A node begins the process by sending a message to all its neighbours and the neighbours, in the next round forward the message to all the neighbours they did not receive the message from and so on. We assume that the nodes do not keep a record of the flooding event. We call this amnesiac flooding (AF). Since the node forgets, if the message is received again in subsequent rounds, it will be forwarded again raising the possibility that the message may be circulated infinitely even on a finite graph. As far as we know, the question of termination for such a flooding process has not been settled - rather, non-termination is implicitly assumed. In this paper, we show that synchronous AF always terminates on any arbitrary finite graph and derive exact termination times which differ sharply in bipartite and non-bipartite graphs. Let $G$ be a finite connected graph. We show that synchronous AF from a single source node terminates on $G$ in $e$ rounds, where $e$ is the eccentricity of the source node, if and only if $G$ is bipartite. For non-bipartite $G$, synchronous AF from a single source terminates in $j$ rounds where $e < j \leq e+d+1$ and $d$ is the diameter of $G$. This limits termination time to at most $d$ and at most $2d + 1$ for bipartite and non-bipartite graphs respectively. If communication/broadcast to all nodes is the motivation, our results show that AF is asymptotically time optimal and obviates the need for construction and maintenance of spanning structures like spanning trees. The clear separation in the termination times of bipartite and non-bipartite graphs also suggests mechanisms for distributed discovery of the topology/distances in arbitrary graphs. For comparison, we show that, in asynchronous networks, an adaptive adversary can force AF to be non-terminating.