In this paper, we introduce a new generalization of the perfect numbers, called -perfect numbers. Briefly stated, an -perfect number is an integer equal to a weighted sum of its proper divisors, where the weights are drawn from some fixed set of integers. After a short exposition of the definitions and some basic results, we present our preliminary investigations into the -perfect numbers for various special sets of small cardinality. In particular, we show that there are infinitely many -perfect numbers and -perfect numbers for every . We also provide a characterization of the -perfect numbers of the form (, an odd prime), as well as a characterization of all even -perfect numbers.