  In the present Colloquium, we focus on the properties of 1-D magnetic systems
on solid surfaces. From the emulation of 1-D quantum phases to the potential
realization of Majorana edge states, spin chains are unique systems to study.
The advent of scanning tunnelling microscope (STM) based techniques has
permitted us to engineer spin chains in an atom-by-atom fashion via atom
manipulation and to access their spin states on the ultimate atomic scale.
Here, we present the current state of research on spin correlations and
dynamics of atomic spin chains as studied by the STM. After a brief review of
the main properties of spin chains on solid surfaces, we classify spin chains
according to the coupling of their magnetic moments with the holding substrate.
This classification scheme takes into account that the nature and lifetimes of
the spin-chain excitation intrinsically depend on the holding substrate. We
first show the interest of using insulating layers on metals, which generally
results in an increase in the spin state's lifetimes such that their quantized
nature gets evident and they are individually accessible. Next, we show that
the use of semiconductor substrates promises additional control through the
tunable electron density via doping. When the coupling to the substrate is
increased for spin chains on metals, the substrate conduction electron mediated
interactions can lead to emergent exotic phases of the coupled spin
chain-substrate conduction electron system. A particularly interesting example
is furnished by superconductors. Magnetic impurities induce states in the
superconducting gap. Due to the extended nature of the spin chain, the in-gap
states develop into bands that can lead to the emergence of 1-D topological
superconductivity and, consequently to the appearance of Majorana edge states.
