This paper proposes a new trust region approach for minimizing a nonlinear function subject to simple bounds. The method does not require solving a quadratic programming subproblem with linear inequalities to obtain an improved step, instead defining a solution to a trust region subproblem by minimizing a quadratic function subject to an ellipsoidal constraint. The iterates generated by this method are always strictly feasible. The proposed methods reduce to a standard trust region approach for unconstrained problems when there are no upper or lower bounds on the variables. Global and quadratic convergence of the methods are established, and preliminary numerical experiments are reported. The paper motivates the method, establishes its convergence properties, and presents two trust region algorithms for bound-constrained problems. The first algorithm, called the double-trust region method, is theoretically interesting and illustrates the essential ideas. The second method is more practical and efficient. The paper also discusses the assumptions required for the convergence analysis and provides detailed proofs of the convergence results.This paper proposes a new trust region approach for minimizing a nonlinear function subject to simple bounds. The method does not require solving a quadratic programming subproblem with linear inequalities to obtain an improved step, instead defining a solution to a trust region subproblem by minimizing a quadratic function subject to an ellipsoidal constraint. The iterates generated by this method are always strictly feasible. The proposed methods reduce to a standard trust region approach for unconstrained problems when there are no upper or lower bounds on the variables. Global and quadratic convergence of the methods are established, and preliminary numerical experiments are reported. The paper motivates the method, establishes its convergence properties, and presents two trust region algorithms for bound-constrained problems. The first algorithm, called the double-trust region method, is theoretically interesting and illustrates the essential ideas. The second method is more practical and efficient. The paper also discusses the assumptions required for the convergence analysis and provides detailed proofs of the convergence results.