Cracking-induced damage and fracture are the most commonly encountered failure modes of engineering materials and structures. In order to prevent such failure, it is a prerequisite in structural designs to understand how cracks nucleate, propagate, branch, coalesces and even fragmentation, etc., in solids, and more importantly, to quantify their adverse effects to the loss of integrity and even catastrophic collapse of structures. Aiming to provide a feasible approach in the modeling of damage and quasi-brittle failure in solids, this work presents systematically the theoretical and numerical aspects of the unified phase-field theory proposed recently by the author, with applications to a couple of representative benchmark problems. Being a variational approach for regularized cracks, this theory incorporates intrinsically the strength-based nucleation and energy-based propagation criteria, as well as the energy minimization-oriented path following criterion, in a standalone framework. Not only several popular phase-field models for brittle fracture can be recovered as particular examples, but also a novel model——the phase-field regularized cohesive zone model (or shortly, PF-CZM)——that applies to both brittle fracture and quasi-brittle failure, emerges naturally. This model can be numerically implemented in context of the coupled finite element method. In order to solve efficiently the discretized governing equations, several numerical algorithms are discussed, with the monolithic BFGS quasi-newton method being the most efficient one. Representative two- and three-dimensional numerical examples reveal that the PF-CZM is capable of reproducing complex fracture configurations in both brittle and quasi-brittle solids under quasi-static, dynamic and multi-physical environments. Remarkably, in all cases objective numerical predictions are achieved independent of the incorporated length scale and mesh discretization. Therefore, the PF-CZM can be used as a numerically predictive approach for the modeling of damage and failure in engineering structures. Finally, some research topics deserving further studies are suggested.