Climate change is a major challenge for modern societies, especially when living in a time of the imperative to constantly increase production. Climate change already affects national economies and society itself, and will continue to do so, since "business as usual" is still playing a great role in the world we live in. This imposes the need to examine the correlation between economic growth and energy consumption. Economic growth generally implies using more non-renewable resources. Therefore, economic growth is mainly associated with adverse environmental impacts, such as exploitation and environmental pollution. However, more recent empirical evidence and data suggest the opposite, i.e. that more developed countries use less energy such as oil, natural gas, and coal, meaning that economic growth can be decoupled or free from adverse environmental impacts. In fact, the economy can grow without using more non-renewable resources and the ensuing environmental problems. In this way, decoupling involves the opposite movements of GDP and renewable energy consumption. In this article, we explore decoupling in the European Union by calculating the decoupling factor. We further tested the beta-convergence hypothesis and conducted a comparative analysis of economic, pollution and decoupling convergence among EU Member States. According to the neoclassical growth theory, poor countries tend to grow faster than rich countries. This idea of convergence can be applied to other fields of (economic) science. Considering that the EU is often regarded of in the context of a two-speed Europe, the aim is to examine how and in which way developed and less developed EU countries relate to "decoupling". Therefore, we grouped the EU countries into three clusters to examine geographical patterns of economic, pollution and decoupling convergence. The convergence analysis and calculation of the decoupling factor was based on per capita GDP and gas emission data. Our results suggested that convergence is happening in the EU in terms of economic and polluting catching-up.