Various block preconditioners for two by two block linear saddle point systems are studied. All block preconditioners are derived from a splitting of the (1,1) block of the two by two block matrix. We analyze the properties of the corresponding preconditioned matrices, in particular their spectra, and discuss the computational performance of the preconditioned iterative methods. It is shown that fast convergence depends mainly on the quality of the splitting of the (1,1) block. Moreover, some strategies of the implementation of the block preconditioners based on purely algebraic considerations are discussed. Thus, applying our block preconditioners to the related saddle point problems, we obtain preconditioned iterative methods in a “black box” fashion.