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The concurrent use of transcranial magnetic stimulation with electroencephalography (TMS-EEG) is growing in popularity as a method for assessing various cortical properties such as excitability, oscillations and connectivity. However, this combination of methods is technically challenging, resulting in artifacts both during recording and following typical EEG analysis methods, which can distort the underlying neural signal. In this article, we review the causes of artifacts in EEG recordings resulting from TMS, as well as artifacts introduced during analysis (e.g. as the result of filtering over high-frequency, large amplitude artifacts). We then discuss methods for removing artifacts, and ways of designing pipelines to minimise analysis-related artifacts. Finally, we introduce the TMS-EEG signal analyser (TESA), an open-source extension for EEGLAB, which includes functions that are specific for TMS-EEG analysis, such as removing and interpolating the TMS pulse artifact, removing and minimising TMS-evoked muscle activity, and analysing TMS-evoked potentials. The aims of TESA are to provide users with easy access to current TMS-EEG analysis methods and to encourage direct comparisons of these methods and pipelines. It is hoped that providing open-source functions will aid in both improving and standardising analysis across the field of TMS-EEG research.