Kolmogorov-Levy-Mandelbrot (t−s)2a-fractional Brownian motion (FBM) appears to be quite relevant for modelling long range memory stochastic systems, and the problem of defining stochastic differential equations subject to such a noise is considered. The Liouville fractional derivative and the self-similarity property of FBM are recalled and then, via detailed calculation, the main statistical characteristics of FBM are derived. First-order stochastic differential equations with FBM are considered via path integrals and a corresponding mean squares approach to non-linear filtering is described. Lastly, a new modelling via stochastic differential equations of fractional order is suggested.