The first part covers the elastoplastic response of the single degree of freedom oscillator. The calculation of the maximum probable displacement of the oscillator is based either on the empirical method of the displacement coefficient or on the capacity spectrum method derived from the hysteretic damping of the system. The results are compared with those of the timehistory analysis of earthquake recordings of the last twenty years in Greece. For stiff, low strength structures, unfavorable soil conditions the results seem to be much higher than the theoretical values therefore improved solutions for practice are proposed. Application of pushover method in multi degree of freedom systems is analyzed followed by a presentation of evaluation strategies, design or redesign of the structure through the increase of stiffness, strength and ductility. Simple relations for the design of earthquake resistant walls are also proposed, suitable for preliminary studies.