Empirical work on regional growth under spatial spillovers uses two workhorse models: the spatial Solow model and Verdoorn's model. This paper contrasts these two views on regional growth processes and demonstrates that in a spatial setting the speed of convergence is heterogenous in both considered models, depending on the remoteness and the income gap of all regions. Furthermore, the paper introduces Wald tests for conditional spatial sigma-convergence based on a spatial maximum likelihood approach. Empirical estimates for 212 European regions covering the period 1980-2002 reveal a slow speed of convergence of about 0.7 percent per year under both models. However, pronounced heterogeneity in the convergence speed is evident. The Wald tests indicate significant conditional spatial sigma-convergence of about 2 percent per year under the spatial Solow model. Verdoorn's specification points to a smaller and insignificant variance reduction during the considered period.