Environmental DNA (eDNA) sampling is prone to both false positive and false negative errors. We review statistical methods to account for such errors in the analysis of eDNA data, and use simulations to compare the performance of different modelling approaches. Our simulations illustrate that even low false positive rates can produce biased estimates of occupancy and detectability. We further show that removing or classifying single PCR detections in an ad-hoc manner under the suspicion that such records represent false positives, as sometimes advocated in the eDNA literature, also results in biased estimation of occupancy, detectability, and false positive rates. We advocate alternative approaches to account for false positive errors that rely on prior information, or the collection of ancillary detection data at a subset of sites using a sampling method that is not prone to false positive errors. We illustrate the advantages of these approaches over ad-hoc classifications of detections, and provide practical advice and code for fitting these models in maximum likelihood and Bayesian frameworks. Given the severe bias induced by false negative and false positive errors, the methods presented here should be more routinely adopted in eDNA studies. This article is protected by copyright. All rights reserved.