The famous Borsuk-Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal point to the same point. A number of important results in combinatorics and discrete geometry have been proved by using this theorem. In this talk we will discuss a generalization of the Borsuk-Ulam type theorem for manifolds. In particular, necessary and sufficient conditions are considered for a manifold with a free involution to be a Borsuk-Ulam type.