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Exactly solvable vertex models might be considered by some physicists as useful toys to conceptualize interesting and elusive effects, but too simple to describe real materials. Instead, the seminal works of Elliott Lieb and others on vertex models have inspired and now sustain the field of artificial spin ices, which produces very concrete artificial nanomagnets hosting phenomena often not seen in natural materials. Here we recall how vertex models were among the inspirations for early artificial spin ices, and how those nanomagnetic realizations in turn motivated further reflections on existing models.We then review how that set of ideas was extended by frustrating the vertex models themselves, to create a veritable zoology of new systems of broadly diverse emergent behaviors. Of these, we review three cases, which were realized and subjected to experimentation: Shakti, tetris, and Santa Fe spin ice.