Quite different search heuristics make use of the concept of assigning an age to search points and systematically remove search points that are too old from the search process. In evolutionary computation one defines some finite maximal lifespan and assigns age 0 to each new search point. In artificial immune systems static pure aging is used. There a finite maximal lifespan is defined but new search points inherit the age of their origin if they do not excel in function value. Both aging mechanisms are supposed to increase the capabilities of the respective search heuristics. A rigorous analysis for two typical difficult situations sheds light on similarities and differences. Considering the behavior on plateaus of constant function values and in local optima both methods are shown to have their strengths and weaknesses. A third aging operator is introduced that provably shares the advantages of both aging mechanisms. Experimental supplements are provided to point out practical implications of the theoretical results and discuss further issues concerning the considered aging strategies.