The mitzva to wipe out the nation of Amalek is primarily incumbent upon the Jewish people as a whole. According to the Sages, the Jewish people were commanded to fulfill three mitzvot upon entering the Land of Israel: appoint a king, wipe out the descendants of Amalek, and build the Holy Temple.
Once the Jews established themselves in the Land with Saul as their king, the prophet Samuel commanded them to destroy the nation of Amalek. Unfortunately, King Saul did not fulfill the mitzva properly. In his war with the Amalekites, he did not chase down all their regiments, and having pity on King Agag, he spared him (as well as the best of the sheep and cattle). As a result, God took the kingdom away from Saul and gave it to David. Nevertheless, devastating damage had been done. Because of Saul’s misplaced compassion, many Amalekites survived and continued harassing the Jews. In fact, there is a tradition that Haman, who wished “to destroy, massacre, and exterminate all the Jews” (Esther 3:13), descended from those very Amalekites.
Today, nobody knows who is descended from Amalek. No one traces their lineage to the Amalekites, or vows to continue in their ways. However, the mitzvot to remember and not forget what Amalek did to us remain in force, as does the mitzva to wipe out Amalek, in the sense of waging an all-out war against bloodthirsty societies that dedicate themselves to murder and terror.