Second-order methods are widely recognized for their rapid convergence; however, this advantage comes at the cost of computing and inverting full Hessians, which becomes impractical at scale. Quasi-Newton (QN) methods address this limitation by using it- erative Hessian approximations, combining strong practical performance with superlin- ear local convergence guarantees. This thesis develops globally convergent Quasi-Newton frameworks that preserve the efficiency of QN updates while providing non-asymptotic global convergence guarantees in convex optimization, variational inequalities, and feder- ated learning. Chapter 2 introduces cubic regularization as a globalization mechanism for Quasi- Newton methods. We analyze inexact second-order models in which the true Hessian is replaced by a QN approximation. By coupling classical limited-memory updates such as L-BFGS with cubic regularization, we obtain global convergence with worst-case iteration complexity matching gradient descent, while approaching the behavior of cubically regu- larized Newton steps when the approximation is accurate. We also propose an efficient algorithm for solving the resulting cubic-regularized subproblem. Chapter 3 proposes the Cubically Enhanced Quasi-Newton (CEQN) step: an explicit stepsize for standard QN directions derived from a cubically regularized model with regu- larization measured in the Hessian approximation norm. CEQN preserves the QN direction while adapting its stepsize to model accuracy, yielding non-asymptotic global convergence in the convex setting. An adaptive variant further modulates the stepsize based on local curvature and approximation error. Chapter 4 develops VIQA, a QN method for solving monotone variational inequali- ties. Jacobians are approximated via low-rank Broyden-type updates, and the resulting subproblems are solved efficiently using Woodbury identities. Under standard smoothness assumptions, we obtain sublinear global convergence rates and identify a verifiable inex- actness condition under which VIQA matches the iteration complexity of optimal exact second-order methods. Chapter 5 adapts QN methodology to Federated Learning. We employ Hessian sketch- ing and gradient compression to construct server-side QN updates that avoid storing dense matrices on devices, reduce communication overhead, and retain convergence under real- istic data heterogeneity. Overall, the thesis advances a unified perspective: approximating second-order informa- tion via Quasi-Newton updates combined with globalization mechanisms yields robust per- formance even from poor initializations. Experiments across multiple benchmarks demon- strate the practical effectiveness of the proposed algorithms, while the theoretical results provide global convergence guarantees.