The assumptions behind Poisson regression are similar to the ones we have for other types of generalised linear models. In addition, we also assume that there is no overdispersion or zero inflation.

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Checklist

Count outcome	The y-variable has to be a count.
Independence of errors	Data should be independent, i.e. not derived from any dependent samples design, e.g. before-after measurements/paired samples.
Correct model specification	Your model should be correctly specified. This means that the x-variables that are included should be meaningful and contribute to the model. No important (confounding) variables should be omitted (often referred to as omitted variable bias).
No multicollinearity	Multicollinearity may occur when two or more x-variables that are included simultaneously in the model are strongly correlated with each another. Actually, this does not violate the assumptions, but is does create greater standard errors which makes it harder to reject the null hypothesis.
No overdispersion	The mean should be equivalent to the variance.
No zero inflation	The difference between observed zeros and predicted zeros is small.

Types of model diagnostics

Link test	Assess model specification
Correlation matrix	Check for multicollinearity
Deviance goodness-of-fit test and Pearson goodness-of-fit test	Assess model fit (no overdispersion or zero inflation)