We introduce and study the general setting of weighted search in which a number of targets, each with a certain weight, are hidden in a star-like environment that consists of $m$ infinite, concurrent rays, with a common origin. A mobile searcher, initially located at the origin, explores this environment so as to locate a set of targets whose aggregate weight is at least a given value $W$. The cost of the search strategy is defined as the total distance traversed by the searcher, and its performance is measured by the worst-case ratio of the cost incurred by the searcher over the cost of an on optimal, offline strategy with complete access to the instance. This is the first study of a setting that generalizes several problems in search theory: the problem in which only a single target is sought, as well as the problem in which all targets have unit weights. We present and analyze a search strategy of near-optimal performance for the problem at hand. We observe that the classical approaches that rely on geometrically increasing search depths perform rather poorly in the context of weighted search. We bypass this problem by using a strategy that modifies the search depths adaptively, depending on the number of targets located up to the current point in time.