Queueing networks are notoriously difficult to analyze sans both Markovian and stationarity assumptions. Much of the theoretical contribution towards performance analysis of time-inhomogeneous single class queueing networks has focused on Markovian networks, with the recent exception of work in Liu and Whitt (2011) and Mandelbaum and Ramanan (2010). In this paper, we introduce transitory queueing networks as a model of inhomogeneous queueing networks, where a large, but finite, number of jobs arrive at queues in the network over a fixed time horizon. The queues offer FIFO service, and we assume that the service rate can be time-varying. The non-Markovian dynamics of this model complicate the analysis of network performance metrics, necessitating approximations. In this paper we develop fluid and diffusion approximations to the number-in-system performance metric by scaling up the number of external arrivals to each queue, following Honnappa et al. (2014). We also discuss the implications for bottleneck detection in tandem queueing networks.