This article discusses the challenges of Higher Education – Brazilian grounds on Demo and Bourdieu relating in particular to the University of the 21st century. Displays the operation of this level of education, so dialectic, in view of the bottlenecks that hamper do research and education in Brazil, especially, that have positive results for society highlighting the social commitment from their real purposes. Part a bibliographic theoretical contribution, whose objective analysis to draw the attention of managers, academics and other professionals of higher education, to the urgent need to think about the higher education of the 21st century, the role of teachers and the University's profile, given the assumptions of Critical Paradigm. It is important to discuss the direction of Brazilian universities and the training of new professionals. Invest in research and education is to invest in development. Hence the confrontation of the scientific community in ensuring the existence of a deliberative Council with the participation of all national scientific entities, to define the allocation of resources and the direction that the University will take in this third millennium. The research that reveal social engagement with his surroundings to institutionalize in an appropriate manner requires connection with education and with the extension.