In this presentation, the purpose is on the development of a fully monolithic solution algorithm for quasi-static phase-field fracture propagation. Phase-field fracture is a variational approach to fracture and consists of two coupled partial differential equations and it is well known that the underlying energy functional is non-convex and sophisticated algorithms are required. The incremental, spatially-discretized problem is treated with adaptive finite elements and predictor-corrector mesh adaptivity that allows for a very small regularization parameter in the crack region. The nonlinear problem is solved with adaptive modified Newton algorithms, which work as inner loop within an inexact augmented Lagrangian iteration for relaxing the crack irreversibility constraint. Specifically, the fully monolithic approach is compared to a quasi-monolithic approach in which phase-field is approximated through extrapolation in the displacement equation. These comparisons are done in terms of certain quantities of interest and computational cost. Moreover, features such as crack nucleation, joining, branching and fracture networks are addressed. All findings are critically commented pointing to open questions and future improvements.