The studies of the emergence of magnetic flux across the convection zone show how the individual magnetic tubes rise toward the surface at the same time remaining anchored at the interface between the convection zone and the radiative interior. A number of recent numerical simulations, in particular, reproduce several observed features of the resulting active regions (e.g., tilt angles, preceding-following asymmetries, etc). These simulations are based on the thin flux tube approximation, which simplifies the magnetic ropes as a one-dimensional continuum. The thin flux tube approximation has provided valuable insights concerning the rise of the magnetic tubes. Yet, it cannot describe some important phenomena occurring during the journey of the tubes across the convection zone. This review summarizes five of those processes. One of them is related to the dramatic off-axis expansion suffered by the top of some magnetic tubes of moderate field strength in the middle of the convection zone (a phenomenon dubbed explosion) as a consequence of the adiabatic character of their evolution. Another one concerns the expansion that all rising tubes experience in the final few $10,000$ km below the photosphere. Further, the thin flux tube approximation does not account for the development of vorticity and twist in the magnetic tubes (or only under very restrictive circumstances). However, vorticity and twist are fundamental ingredients that have to be considered at various stages of the rise. Without twist, the buoyant tubes tend to split and yield vortex filament pairs which separate horizontally instead of rising. The transverse field component of a twisted tube helps maintain the unity of the tube, but it yields an evolutionary pattern far more complicated than as described by the thin flux tube approximation. All this is explained on the basis of the recent results of two-dimensional MHD simulations of the initial stages of the rise. Finally, the back reaction of the external medium to the advance of the magnetic region is difficult to treat in the thin tube approximation. The simple local prescription commonly used is inconsistent in that it is based on the assumption of a potential (i.e., essentially non-local) flow around the tube. As a result, the expression for the enhanced inertia it yields may violate the condition of global momentum conservation. In the final section of the paper, the validity of the local approach for a rectilinear and non-rotating tube as well as its failure in more general cases are explained.