In the presence of strong density stratification, hydromagnetic turbulence attains qualitatively new properties: the formation of magnetic flux concentrations. We review here the theoretical foundations of this mechanism in terms of what is now called the negative effective magnetic pressure instability. We also present direct numerical simulations of forced turbulence in strongly stratified layers and discuss the qualitative and quantitative similarities with corresponding mean-field simulations. Finally, the relevance to sunspot formation is discussed.Numerical simulations with realistic surface physics have successfully produced active region formation from an unstructured initial magnetic field [27], but it is still a large leap to modeling actual sunspots [28].Meanwhile, several simulations have displayed spontaneous magnetic structure formation. Some of them involve turbulent convection [16,27,[29][30][31], or a stably stratified polytropic atmosphere [32], so it remains to be clarified, whether gravity plays a direct role, or whether the magnetic field concentrations are mainly the result of converging downdrafts. Other simulations involve forced turbulence in isothermally stratified domains, where no thermally driven convection is possible [10][11][12][13][14][33][34][35], and yet one finds the formation of large-scale magnetic structures. What is remarkable is that these structures extend over the scale of many turbulent eddies. This property suggests that they should be amenable to a mean-field treatment involving averaged, effective equations.A mean-field approach relevant for describing magnetic effects on the mean flow was developed nearly three decades ago [4][5][6][7][8][9], but only in recent years, with the assistance of DNS, has it gained sufficient attention. In the following, we review the essential properties of NEMPI, discuss analytical approaches to the understanding of the behavior in the presence of either horizontal or vertical magnetic fields, and then turn to DNS whose results can be understood in terms of NEMPI.