We study harmonic maps from an admissible flat simplicial complex to a non-positively curved Riemannian manifold. Our
main regularity theorem is that these maps are C1, at the interfaces of the top-dimensional simplices in addition to satisfying a
balancing condition. If we assume that the domain is a 2-complex,then these maps are C1. As an application, we show that the
regularity, the balancing condition and a Bochner formula lead to rigidity and vanishing theorems for harmonic maps. Furthermore,
we give an explicit relationship between our techniques and those obtained via combinatorial methods.