n-fold categories are obtained by n-iterations of the internal category construction, starting from the category Cat. N-Fold categories satisfying the globularity condition correspond to strict n-categories. One can ask if there is an intermediate structure between n-fold categories and strict n-categories which is a model of weak n-categories. We answer this question positively with the introduction of the category of weakly globular n-fold categories. This category is based on a new paradigm to weaken higher categorical structures, and offers several advantages in terms of applications. We will illustrate how this model is part of a wider class of Segal-type models, and its equivalence with the Tamsamani-Simpson model of weak n-categories.