The problem of the demonstration of bicontinuous structures in microemulsions is critically examined, with an emphasis on systems with low concentrations of surfactant (s) and comparable volume fractions of oil and water. While the problem of microemulsion bicontinuity has been investigated by a large number of experimental approaches, it is found that to date, only a few can provide an unambiguous clarification of this point. Multicomponent self-diffusion data appear to provide the most convenient and reliable examination of the problem, and therefore the possibilities and limitations of this approach are analysed in some detail. From experimental studies it is found that microemulsions with approximately equal volume fractions of the two solvents are in general bicontinuous, also in the most narrow sense of the word. It is also demonstrated that surfactant molecules in these microemulsions are arranged in monomolecular layers, which are characterized by a spontaneous packing into surfaces of zero, or at least low, mean curvature. The relation between surfactant packing characteristics and microemulsion structure is discussed, in particular for the typically observed sequence of curvature towards oil ⪢ zero mean curvature ⪢ curvature towards water, on change of some parameter (e.g., salinity, cosurfactant, surfactant composition, temperature). As regards an understanding of the physical mechanisms behind alterations in surfactant packing (and thus also microstructure), the problem of nonionic oligo (ethylene oxide) surfactants appears to provide the most intriguing (and controversial) problem. This problem is therefore discussed in a slightly broader context.