The structure, dynamics and stability of Fe-Pt nanoparticles have been investigated using DFT-based techniques: total energy calculations and molecular dynamics. The investigated systems included multi-shell and disordered nanoparticles of iron and platinum. The study concerns icosahedral particles with the magic number of atoms (55): iron-terminated Fe43Pt12, platinum-terminated Fe12Pt43, and disordered Fe27Pt28. Additionally, the Fe6Pt7 cluster has been investigated to probe the behaviour of extremely small Fe-Pt particles. Molecular dynamics simulations have been performed for a few temperatures between T = 150-1000 K. The calculations revealed high structural instability of the Fe-terminated nanoparticles and a strong stabilising effect of the Pt-termination in the shell-type icosahedral particles. The platinum termination prevented disordering of the particle even at T = 1000 K indicating very high melting temperatures of these Fe-Pt icosahedral structures. The analysis of evolution of the radial distribution function has shown a significant tendency of Pt atoms to move to the outside layer of the particles - even in the platinum deficient cases.