We develop spatial statistical models for stream networks that can estimate relationships between a response variable and other covariates, make predictions at unsampled locations, and predict an average or total for a stream or a stream segment. There have been very few attempts to develop valid spatial covariance models that incorporate flow, stream distance, or both. The application of typical spatial autocovariance functions based on Euclidean distance, such as the spherical covariance model, are not valid when using stream distance. In this paper we develop a class of valid models that incorporate flow and stream distance by using spatial moving averages. These methods integrate a moving average function, or kernel, against a white noise process. We use restricted maximum likelihood to fit a valid covariance matrix that uses flow and/or stream distance, and then we use this covariance matrix to estimate fixed effects and make kriging and block kriging predictions. Using an example data set from the Maryland Biological Stream Survey (MBSS), we show that models using flow for sulfate concentrations are more appropriate than models that only use stream distance. For some fish counts, pure stream distance models are most appropriate.