The initial and significant step in the design of a cellular manufacturing system is cell formation (CF). CF problem is proposed in this paper as a decision problem that determines to manufacture specified types of part in a manufacturing plant which machines and their associated parts are grouped together to form cell in a way that a concerned objective is optimized. For describing CF problem clearly, this paper firstly presents a review of cell formation problem from the view points of objective function. The CF problems are classified into three categories, which are cost oriented, flexibility oriented and grouping efficiency oriented CF problems. Then, the paper presents a comprehensive conceptual mathematical formulation describing the general cost problem and a decision variable for comprehensive describing routing flexibility and two trade-off questions in grouping efficiency issues. Finally, based on the review and discussion, the paper proposes five directions for future research in the CF field.