It is important to model biological variation when analyzing spatial transcriptomics data from multiple samples. One approach to multi-sample analysis is to spatially align samples, but this is a challenging problem. Here, we provide an alignment-free framework for generalizing a one-sample spatial factorization model to multi-sample data. Using this framework, we develop a method, called multi-sample non-negative spatial factorization (mNSF) that extends the one-sample non-negative spatial factorization (NSF) framework to a multi-sample dataset. Our model allows for a sample-specific model for the spatial correlation structure and extracts a low-dimensional representation of the data. We illustrate the performance of mNSF by simulation studies and real data. mNSF identifies true factors in simulated data, identifies shared anatomical regions across samples in real data and reveals region-specific biological functions. mNSFs performance is similar to alignment based methods when alignment is possible, but extends analysis to situations where spatial alignment is impossible. We expect multi-sample factorization methods to be a powerful class of methods for analyzing spatially resolved transcriptomics data.