Lowness has been studied in the contexts of degree theory, learning theory, and randomness. I will discuss lowness in the context of recursive model theory: we say that a degree is low for isomorphism if, whenever it can compute an isomorphism between two recursively presented structures, there is actually a recursive isomorphism between them. I will describe the class of Turing degrees that are low for isomorphism, identify some particular subclasses, and show how it behaves with respect to measure and category.