In this paper, we study causal discrete memoryless relay networks. The network consists of multiple nodes, each of which can be a source, a relay, and/or a destination. In the network, there are two types of relays: 1) relays with one sample delay (strictly causal) and 2) relays without delay (causal) whose transmit signals depend not only on the past received symbols but also on the current received symbols. For this network, we derive two new cut-set bounds, one when every node has a message and the other when only the strictly causal relays have messages. Using the examples of a causal vector Gaussian two-way relay channel and a causal vector Gaussian relay channel, we show that the new cut-set bounds can be achieved by a simple amplify-and-forward type relaying. Our result for the causal relay channel strengthens the previously known capacity result for the same channel by El Gamal, Hassanpour, and Mammen.