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Downscaling is a landmark task in climate science and meteorology in which the goal is to use coarse scale, spatio-temporal data to infer values at finer scales. Statistical downscaling aims to approximate this task using statistical patterns gleaned from an existing dataset of downscaled values, often obtained from observations or physical models. In this work, we investigate the application of deep latent variable learning to the task of statistical downscaling. We present ClimAlign, a novel method for unsupervised, generative downscaling using adaptations of recent work in normalizing flows for variational inference. We evaluate the viability of our method using several different metrics on two datasets consisting of daily temperature and precipitation values gridded at low (1 degree latitude/longitude) and high (1/4 and 1/8 degree) resolutions. We show that our method achieves comparable predictive performance to existing supervised statistical downscaling methods while simultaneously allowing for both conditional and unconditional sampling from the joint distribution over high and low resolution spatial fields. We provide publicly accessible implementations of our method, as well as the baselines used for comparison, on GitHub.