This work addresses the description problem of a target class in the presence of negative samples or outliers. Traditional Support Vector Machines (SVM) has strong discrimination capability to distinguish the target class but does not reject the uncharacteristic patterns well. The one-class SVM, on the other hand, provides good representation for the class of interest but overlooks the discrimination issue between the class and outliers. This paper presents a new one-class classifier named minimum enclosing and maximum excluding machine (MEMEM), which offers capabilities for both pattern description and discrimination. The properties of MEMEM are analyzed and the performance comparisons using synthetic and real data are presented.