Understanding the physical processes that control star formation in the interstellar medium (ISM) is one of the key themes in the astrophysics of galaxies today. In the current paradigm of turbulence-regulated ISM, star formation is intricately linked to the internal structure of individual molecular clouds. This internal structure is a result of a complex interplay of turbulence, gravity, and magnetic fields in the clouds. The fundamental open question is that we do not know how exactly these processes affect the ISM structure, giving rise to its characteristic properties. In this talk, I present our recent efforts to confront the turbulence-regulated ISM paradigm using the density structure of the cold ISM as an observational diagnostic. I show how this approach has led to new constraints for some of the most fundamental measures of the molecular cloud structure. I will also discuss the role of filaments as a part of the molecular cloud structure. I focus on our recent results regarding filament fragmentation and the challenges those results pose to our theoretical understanding of filament evolution.