This project aims at exploiting the wide-field and limiting-magnitude capabilities of the LSST to fully characterise the resolved stellar populations in/around six Local Group stellar systems of different morphological type that are located from \sim 30 to \sim 400 kpc in distance from us . The adopted stellar tracers will be mainly red giants , pulsating variable stars of different types , and turn-off ( TO ) main sequence ( MS ) stars . In a synergistic and complemental approach , we have selected targets that host red giant branch ( RGB ) stars which are within the reach of Gaia and not yet ( all ) saturated with the LSST . In five of them we will reach at least one magnitude below the TO of the oldest stellar component ( t \geq 10 Gyr ) with the LSST . Specifically , we plan to use pulsating variable stars populating the whole classical instability strip ( namely : RR Lyrae stars , Cepheids of different types , SX Phoenicis and delta Scuti stars ) and Long Period Variables ( LPVs ) , along with the Color Magnitude Diagram ( CMD ) of the resolved stellar populations in these 6 systems to : i ) trace their different stellar generations over the large spatial extension and magnitude depth allowed by the LSST ; ii ) measure their distances using variable stars of different type/parent stellar population and the Tip of the Red Giant Branch ( TRGB ) ; iii ) map their 3D structures all the way through to the farther periphery of their halos ; iv ) search for tidal streams which are supposed to connect some of these systems ; and v ) study their Star Formation Histories ( SFHs ) over an unprecedented large fraction of their bodies . Our ultimate goal is : i ) to provide a complete picture of these nearby stellar systems all the way through to their periphery , and : ii ) to directly link and cross-calibrate the Gaia and LSST projects . A valuable by-product will be the derivation , for the first time , of period-luminosity relations based on statistically significant samples of delta Scuti and SX Phoenicis stars , in different environments .