Summary: The class of moving-average fractional Lévy motions (MAFLMs), which are fields parameterized by a \(d\)-dimensional space, is introduced. MAFLMs are defined by a moving-average fractional integration of order \(H\) of a random Lévy measure with finite moments. MAFLMs are centred \(d\)-dimensional motions with stationary increments, and have the same covariance function as fractional Brownian motions. They have \(H-d/2\) Hölder continuous sample paths. When the Lévy measure is the truncated random stable measure of index \(\alpha\), MAFLMs are locally self-similar with index \(\widetilde{H}= H-d/2+ d/\alpha\). This shows that in a non-Gaussian setting these indices (local self-similarity, variance of the increments, Hölder continuity) may be different. Moreover, we can establish a multiscale behaviour of some of these fields. All the indices of such MAFLMs are identified for the truncated random stable measure.