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Aim: Invasions of non‐native insects can have substantial impacts on agriculture, forestry, human health and biodiversity with considerable economic and environmental consequences. To understand the causes of these invasions, it is important to quantify the relative influence of principal drivers such as international imports and climatic effects. Location: North America, Chile, Europe, Australia, New Zealand and Japan. Time Period: 1881–2020. Methods: To evaluate the relative contributions of various factors in explaining global variation in numbers of non‐native insect establishments in different world regions, we conducted two multivariate regression analyses to quantify temporal changes in family‐level composition and native ranges of established non‐native species in several world regions. Results: There were temporal changes in the family‐level composition of non‐native species assemblages. Prior to 1900, invasions were dominated by scale insects, subsequently shifting to a more diverse set of species, except in North America, which had relatively small compositional change over time compared to other regions. Spatial and temporal variation in the composition of established species was associated with differences in the origin of imports and climatic factors, each explaining 26.3% and 27.4% of the total variation, respectively. The analysis of native ranges of non‐native species indicated that there was no consistent temporal variation across all regions. Established species in New Zealand were predominantly native to Australasia and species in North America and Chile were mainly from Europe. Non‐native species in Europe mainly originated from the Nearctic region while those in Japan and Australia generally originated from multiple regions. Climatic factors in the destination regions had a primary effect (66.3%) on variation in the native range of established species, although imports also had substantial effects (45.4%). Main Conclusions: Geographical variation in climate and imports act together as drivers of establishment success for non‐native insects in all six regions. [ABSTRACT FROM AUTHOR]