Zero or null morphology refers to morphological units that are devoid of phonological content. Whether such entities should be postulated is one of the most controversial issues in morphological theory, with disagreements in how the concept should be delimited, what would count as an instance of zero morphology inside a particular theory, and whether such objects should be allowed even as mere analytical instruments.
With respect to the first problem, given that zero morphology is a hypothesis that comes from certain analyses, delimiting what counts as a zero morpheme is not a trivial matter. The concept must be carefully differentiated from others that intuitively also involve situations where there is no overt morphological marking: cumulative morphology, phonological deletion, etc.
About the second issue, what counts as null can also depend on the specific theories where the proposal is made. In the strict sense, zero morphology involves a complete morphosyntactic representation that is associated to zero phonological content, but there are other notions of zero morphology that differ from the one discussed here, such as absolute absence of morphological expression, in addition to specific theory-internal interpretations of what counts as null. Thus, it is also important to consider the different ways in which something can be morphologically silent.
Finally, with respect to the third side of the debate, arguments are made for and against zero morphology, notably from the perspectives of falsifiability, acquisition, and psycholinguistics. Of particular impact is the question of which properties a theory should have in order to block the possibility that zero morphology exists, and conversely the properties that theories that accept zero morphology associate to null morphemes.
An important ingredient in this debate has to do with two empirical domains: zero derivation and paradigmatic uniformity. Ultimately, the plausibility that zero morphemes exist or not depends on the success at accounting for these two empirical patterns in a better way than theories that ban zero morphology.