We present and discuss methodologies for rapid determination of the magnitude and tsunami potential of large earthquakes. Firstly, a duration-amplitude procedure gives an earthquake moment magnitude, M_wpd, within 10-20 min after origin time (OT) using P-wave recordings at teleseismic distances. This procedure determines apparent source durations, T₀, from high-frequency, P-wave records, and then estimates moments through integration of broadband displacement waveforms over the interval t_P to t_P+T₀, where t_P is the P arrival time. We apply the duration-amplitude methodology to a set of recent, large earthquakes (M_w^CMT 6.6 to 9.3) with diverse source types. The results show that a scaling of the moment estimates for larger interplate thrust and possibly tsunami earthquakes is necessary to best match M_w^CMT. With this scaling, M_wpd matches M_w^CMT typically within +/-0.2 magnitude units and does not exhibit saturation, even for the largest events. The explicit use of the source duration for integration of displacement seismograms, the moment scaling, and other characteristics of the duration-amplitude methodology make it an extension of the widely used, M_wp, rapid-magnitude procedure. The need for a moment scaling may be related to the destructive interference of pP or sP waves with direct, down-going P waves; in this case some of the energy released during rupture may be re-absorbed locally to further drive the rupture, and thus to make the earthquake large. Secondly, a direct procedure for rapid assessment (6-10 min after OT) of earthquake tsunami potential uses two, simple measures on P-wave seismograms – the predominant period on velocity records, T_d, and the likelihood that the high-frequency, apparent source duration, T₀, exceeds 50-55 sec. For the set of recent, large earthquakes, the period-duration product T_dT₀ gives more information on tsunami impact and size than M_w^CMT and other currently used discriminants. All discriminants have difficulty in assessing the tsunami potential for strike-slip and back-arc, intraplate earthquakes. T_d and T₀ can be related to the critical parameters length, L, width, W, mean slip, D, and depth, z, of earthquake rupture. Our results suggest that tsunami potential is not directly related to the product LWD (the “seismic” faulting model), as is assumed with the use of the M_w^CMT discriminant. Instead, knowledge of rupture length, L, and depth, z, alone can constrain well the tsunami potential of an earthquake (the “tsunami” faulting model), with explicit determination of fault width, W, and slip, D, being of secondary importance.