In this paper we introduce new notions of k-type anonymizations. Those, notions achieve similar privacy goals as those aimed by Sweenie and Samarati when proposing the concept of k-anonymization: an adversary who knows the public data of an individual cannot link that individual to less than k records in the anonymized table. Every anonymized table that satisfies k-anonymity complies also with the anonymity constraints dictated by the new notions, but the converse is not necessarily true. Thus, those new notions allow generalized tables that may offer higher utility than k-anonymized tables, while still preserving the required privacy constraints. We discuss and compare the new anonymization concepts, which we call (1, k)-. (k, k)- and global (1, k)-anonymizations, according to several utility measures. We propose a collection of agglomerative algorithms for the problem of finding such anonymizations with high utility, and demonstrate the usefulness of our definitions and our algorithms through extensive experimental evaluation on real and synthetic datasets.