Huge linear systems arise in finite element practice and a considerable effort has been expended in developing domain decomposition methods and other fast iterative solvers to find the solutions. In this contribution, we will consider the symmetric indefinite systems that arise in mixed finite element approximations of the equations of linear elasticity and of the incompressible Stokes equation. Our methods are of balancing Neumann-Neumann type. The preconditioner is built from many much smaller saddle point problems representing the problem on the subdomains and a coarse saddle point problem with only a few degrees of freedom for each subdomain. We will present the basic design of our algorithm and theoretical results which show scalability with respect to the number of subdomains and a very modest growth in the iteration numbers as a function of the size of the subdomain problems. Our method has been implemented successfully using the PETSc system developed at Argonne National Laboratories and we will discuss our experience of solving systems with millions of variables on a Beowulf type computer system. Our research is conducted jointly with Luca F. Pavarino of the University of Milan, Italy and Paulo Goldfeld of the Courant Institute.