Rapid assessment of earthquake size and tsunami potential of large earthquakes is of primary importance for issuing warnings and alarms; tsunami monitoring centers throughout the world can benefit from all practical and reliable procedures for this assessment. In general, it is accepted that monitoring systems must rely on a series of analysis tools that stream information progressively, from early, rapid and generally more uncertain estimates, to later, more accurate results. Seismic P-waves are the first seismic data recorded by network stations, these waves carry information on the earthquake source rupture including its extension and depth. For example, the duration of high frequency P-waves can be related to fault rupture length, and the P-wave dominant period can be related to fault width and/or the fault slip. These quantities and their combinations can be determined on-the-fly on the incoming data streams and can be at seismic monitoring centers in near-real time. In this work, we present the procedures implemented at the Istituto Nazionale di Geofisica e Vulcanologia, INGV, for the purpose of global and Euro-Mediterranean scale tsunami warning and alert. These procedures provide rapid measures of earthquake tsunami potential by exploiting both the high-frequency duration and the dominant period of P-waves. These discriminants are complemented by fast earthquake location (5-10 minutes depending on station coverage and source depth) and rapid calculation of the standard Mwp and of the non-saturating Mwpd magnitudes. All this information is updated each minute and displayed using a comprehensive, graphical representation that allows rapid visual inspection and understanding of the status of global earthquake occurrence. Examples drawn from recent important earthquakes will be shown.