This talk is based on my project with Lisa Nicklasson. Let L be a finite distributive lattice and let t_1,...,t_n denote the elements of its ground set. To each maximal chain C of L one can associate a squarefree monomial m in K[t_1,...,t_n] which equals the product of all t_i belonging to C. We then consider the subalgebra K[m_1,...,m_s], generated by all such monomials, and call it the chain algebra of L. In my talk I will discuss some properties of chain algebras in connection to combinatorial properties of the corresponding lattices, and the connection of such algebras to Hibi rings.