We consider inference using multivariate data that are misaligned both in space and in support. Previous work (Mugglin and Carlin, 1998; Mugglin, Carlin, and Gelfand, 1999) considers the analysis of random variables (typically counts or rates) which are aggregated over differing sets of possibly non-nested regional boundaries. This sort of areal misalignment is handled using conditionally independent Poisson-multinomial models, thus offering a Bayesian solution to the celebrated modifiable areal unit problem (MAUP). Explanatory covariates and multilevel responses can also be easily accommodated, with spatial correlation modeled using a conditionally autoregressive (CAR) prior structure. In this talk we extend this approach in three ways. First, we imagine an underlying continuous spatial process Y(s) for locations s in D, a region of interest. This allows us to consider the general change of support problem (COSP), where we seek to make inferences about the values of a variable at either points or regions different from those at which it has been observed. Second, we apply our COSP approach to the spatio-temporal case, and show that the additional computational burden to analyze the correspondingly larger data set still emerges as manageable. Finally, we consider the full misaligned regression setting, where for example the spatially-referenced predictor variable is point process in nature, while the response is only available either as areal summaries over a particular grid. We illustrate all three of these developments using a dataset relating several air quality indicators (ozone, particulate matter, nitrogen oxides, etc.), a range of sociodemographic variables (race, age, gender, race, and a socioeconomic status surrogate), and pediatric emergency room (ER) visit counts for asthma in the Atlanta, Georgia metropolitan area. Here the air quality data is collected at fixed monitoring stations (point processes) while the sociodemographic covariates and response variable is collected by zip code (areal summaries). Like many recent hierarchical Bayesian spatial applications, computing is implemented via a carefully tailored Metropolis-Hastings algorithm, with map summaries created using a geographic information system (GIS). This talk covers joint work with Li Zhu of the Division of Biostatistics at the University of Minnesota, and Alan Gelfand of the Department of Statistics at the University of Connecticut.