Due to progress in nanotechnology high-quality quantum wires can nowadays be fabricated. The behavior of particles in one dimension differs significantly from that in three-dimensional (3D) systems, yet the physics of such low-dimensional systems is generally not very well represented in standard undergraduate or graduate curricula. For instance, the Fermi liquid paradigm, working well for electrons in 3D systems, breaks down in 1D and has to be replaced by the Tomonaga-Luttinger liquid (TLL) description. However, most of the introductory papers on the TLL restrict themselves to a summary of results to be compared with experiments, while review papers are often fraught with technical details only accessible to the theorist. The present paper provides an elementary discussion of some phenomena distinguishing quantum wires and quasi-one-dimensional systems from their 3D counterparts targeting experimentalists and graduate students. It aims to convey the basic ideas of TLL theory and to relate it to alternative phases, such as charge and spin density waves. The important role of fluctuations in quasi-1D systems is pointed out and the connection is made to the problem of high-Tc superconductivity. The discussion is kept on a level, which should be accessible with a basic knowledge of quantum mechanics.