Elementary spin excitations (magnons) play a fundamental role in condensed matter physics, since many phenomena e.g. magnetic ordering, electrical (as well as heat) transport properties, ultrafast magnetization processes, and most importantly electron/spin dynamics can only be understood when these quasi-particles are taken into consideration. In addition to their fundamental importance, magnons may also be used for information processing in modern spintronics. Here the concept of spin excitations in ultrathin itinerant magnets is discussed and reviewed. Starting with a historical introduction, different classes of magnons are introduced. Different theoretical treatments of spin excitations in solids are outlined. Interaction of spin-polarized electrons with a magnetic surface is discussed. It is shown that, based on the quantum mechanical conservation rules, a magnon can only be excited when a minority electron is injected into the system. While the magnon creation process is forbidden by majority electrons, the magnon annihilation process is allowed instead. These fundamental quantum mechanical selection rules, together with the strong interaction of electrons with matter, make the spin-polarized electron spectroscopies as appropriate tools to excite and probe the elementary spin excitations in low-dimensional magnets e.g ultrathin films and nanostructures. The focus is put on the experimental results obtained by spin-polarized electron energy loss spectroscopy and spin-polarized inelastic tunneling spectroscopy. The magnon dispersion relation, lifetime, group and phase velocity measured using these approaches in various ultrathin magnets are discussed in detail. The differences and similarities with respect to the bulk excitations are addressed. The role of the temperature, atomic structure, number of atomic layers, lattice strain, electronic complexes and hybridization at the interfaces are outlined. A possibility of simultaneous probing of magnons and phonons in complex low-dimensional ferromagnetic oxide nanostructures is discussed. The influence of the relativistic spin-orbit coupling on high-energy magnons is addressed. It is shown how the spin-orbit coupling breaks the energy degeneracy of the magnons excited in an ultrathin ferromagnet, and how it influences their lifetime, amplitude, group and phase velocity. A potential application of these new effects in modern spintronics is outlined. It is illustrated how one can take advantage of collective nature of magnons and use these quasi-particles for probing the magnetic exchange interaction at buried interfaces.