Abstract: OLAP queries (i.e. group-by or cube-by queries with aggregation) have proven to be valuable for data analysis and exploration. Many decision support applications need very complex OLAP queries, requiring a fine degree of control over both the group definition and the aggregates that are computed. For example, suppose that the user has access to a data cube whose measure attribute is Sum(Sales). Then the user might wish to compute the sum of sales in New York and the sum of sales in California for those data cube entries in which Sum(Sales) > $1,000,000. This type of complex OLAP query is often difficult to express and difficult to optimize using standard relational operators (including standard aggregation operators). In this paper we propose the MD-join operator for complex OLAP queries. The MD-join provides a clean separation between group definition and aggregate computation, allowing great flexibility in the expression of OLAP queries. In addition, the MD-join has a simple and easily optimizable implementation, while the equivalent relational algebra expression is often complex and difficult to optimize. We present several algebraic transformations that allow relational algebra queries that include MD-joins to be optimized.