A general framework for optiinal stopping problems associated with multivariate point processes is presented. Such problems arise. for example. in finding optimal selection/detection strategies in discovering new species, error detection and job search problems. The optimal stopping problem is solved in a finite and an infinite horizon setting. and various special models are coilsidered. It turns out that in regenerative type of modlels where the interarrival time distribtutions have properties known from reliability theory. such as NBUE, IFR and DFR, the optimal stopping times have appealirig properties.