FSQP is a set of Fortran subroutines for the minimization of a smooth objective function subject to nonlinear smooth inequality constraints, linear inequality and linear equality constraints, and simple bounds on the variables. If the initial guess provided by the user is infeasible, FSQP first generates a feasible point from the given point. Subsequently the successive iterates generated by FSQP all satisfy the constraints. The user also has the option of requiring that the objective value decrease at each iteration after feasibility has been reached. The user must provide subroutines that define the objective and constraint functions and may either provide the subroutines that define the gradients of these functions or require that FSQP estimate them by forward finite differences. FSQP uses an algorithm based on Sequential Quandratic Programming (SQP), modified so as to generate feasible iterates. A certain arc search ensures that the step of one is eventually satisfied, a requirement for superlinear convergence. The merit function used in this arc search is the objective function itself, and either an Armijo- type line search or a nonmonotone line search borrowed from Grippo et al. may be selected.