A finite collection C of k-sets, where is called a k-clique if every two k-sets (called lines) in C have a nonempty intersection and a k-clique is a called a maximal k-clique if and C is maximal with respect to this property. That is, every two lines in C have a nonempty intersection and there does not exist A such that , and for all . An elementary example of a maximal k-clique is furnished by the family of all the k-subsets of a -set. This k-clique will be called the binomial k-clique. This paper is intended to give some combinatorial characterizations of the binomial k-clique as a maximal k-clique. The techniques developed are then used to provide a large number of examples of mutually nonisomorphic maximal k-cliques for a fixed value of k.