We calculated the impact on Southern Italy of a large set of tsunamis resulting from earthquakes generated by major fault zones of the Mediterranean Sea. Our approach merges updated knowledge on the regional tectonic setting and scenario-like calculations of expected tsunami impact. We selected three potential source zones located at short, intermediate and large distance from our target coastlines: the Southern Tyrrhenian thrust belt; the Tell-Atlas thrust belt; and the western Hellenic Arc. For each zone we determined a Maximum Credible Earthquake and described the geometry, kinematics and size of its associated Typical Fault. We then let the Typical Fault float along strike of its parent source zone and simulated all tsunamis it could trigger. Simulations are based on the solution of the nonlinear shallow water equations through a finite difference technique. For each run we calculated the wavefields at desired simulation times and the maximum water elevation field, then produced traveltime maps and maximum wave-height profiles along the target coastlines. The results show a highly variable impact for tsunamis generated by the different source zones. For example, a large Hellenic Arc earthquake will produce a much higher tsunami wave (up to 5 m) than those of the other two source zones (up to 1.5 m). This implies that tsunami scenarios for Mediterranean Sea countries must necessarily be computed at the scale of the entire basin. Our work represents a pilot study for constructing a basin-wide tsunami scenario database to be used for tsunami hazard assessment and early warning.