In this paper we establish a standard monomial theory for generic Hankel matrices. By a generic Hankel matrix we mean a matrix Y=( yij) with yij=x i+ j&1 where the x i are indeterminates over a field K. We use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals It of Y. Further we prove that the symbolic and ordinary Rees algebras of It are Cohen Macaulay normal domains. The first standard monomial theory was developed by Hodge [H] to study the homogeneous coordinate ring of the Grassmannian variety. Later standard monomial theories were established for generic matrices by Doubilet, Rota and Stein [DRS], and for generic symmetric and generic skew symmetric matrices by De Concini and Procesi [DP]. These are all examples of algebras with straightening law (ASL for short) over a poset or over a doset. The abstract notion of ASL was introduced and developed by Eisenbud [E1], and by De Concini, Eisenbud and Procesi [DEP2], see also [BV]. These structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices. Let now x1 , ..., xn be indeterminates over an arbitrary field K. For j=1, ..., n we denote by Xj the j_(n+1& j) Hankel matrix with entries x1 , ..., xn , that is,