Phase-field models are widely regarded as one of the most versatile methods to numerically solve a variety of physical phenomena related to Continuum Mechanics. In this dissertation they are applied to some notable brittle fracture mechanics case studies, in order to validate their robustness and reliability by comparing produced numerical outcomes with experimental results. Firstly a 1D benchmark is defined, giving particular attention to modeling definitions (e.g. how degradation functions depends on damage rise and growth) and stability concepts, investigating under which conditions damage localizes. For this one-dimensional case an original MATLAB code was implemented, whose logic and structure are broadly deepen.