This paper concerns some practical issues associated with the formulation of sequential quadratic programming (SQP) methods for large-scale nonlinear optimization. SQP methods find an approximate solution of a sequence of quadratic programming (QP) subproblems in which a quadratic model of the objective function is minimized subject to the linearized constraints. Extensive numerical results are given for 1153 problems from the CUTEst test collection. The results indicate that SQP methods based on maintaining a quasi-Newton approximation to the Hessian of the Lagrangian function are both reliable and efficient for general large-scale optimization problems. In particular, the results show that in some situations, quasi-Newton methods are more efficient than competing methods based on the exact Hessian of the Lagrangian. The paper concludes with discussion of an SQP method that employs both approximate and exact Hessian information. In this approach the quadratic programming subproblem is either the conventional subproblem defined in terms of a positive-definite quasi-Newton approximate Hessian, or a convexified problem based on the exact Hessian.