The paper presents an analysis of the convergence process of the households' final consumption across European Union countries in the period of 2004 - 2016, considering spatial dependencies. The consumption is the main factor of the Gross Domestic Product. Previous studies focused on the economic convergence - a few researchers analyzed consumption convergence. This paper is based on previous studies on consumption convergence. The primary aim of the study is to verify hypothesis about the occurrence of the convergence process in households' final consumption expenditures across EU countries. The second aim is to verify whether whether the sustainable consumption significantly influences the researched process. The sustainable consumption is one of the several Sustainable Development Goals (SDGs) defined in the 2030 Agenda for Sustainable Development. However, some EU countries that want to align the level of consumption expenditures do not pay attention at sustainability. It depends among others on the consumerism idea. In the analysis, the level of the households' final consumption expenditures per capita is applied as the consumption. The sustainable consumption process is calculated using Hellwig's taxonomic measure of development (TMD), based on four diagnostic variables: final energy consumption, energy productivity, share of renewable energy in gross final energy consumption, resource productivity and domestic material consumption (DMC). To verify spatial dependencies in considered processes the spatial autocorrelation is tested. Consumption convergence process is verified with absolute and conditional beta-convergence approach. In the first place, beta-convergence panel data models are estimated and verified. Then, models are supplemented with a spatial factor - the dependence on the neighborhood countries. The spatial autoregressive and spatial error beta-convergence panel data models are used. In the investigation of the conditional convergence the following indicators are added: the sustainable consumption and the period of financial crisis. The financial crisis slowed down the consumption convergence process in EU countries. As a result of the study, the sustainable consumption does not have a significant influence on the convergence process.