The question is pretty much in the title, but I am really seeking countries which kept slaves around the same time the US did, and where it was just as prominent in said country as it was in the US.
Most African slaves were sent to the Americas — basically a new world with large industrial-scale labor-intensive agriculture. There isn't much point in taking slaves to Europe to replace serfs or cheap farm laborers on small farms.
Destinations of slaves Africa→Americas
Portuguese America (modern Brazil) 38.5% British America (minus North America) 18.4% Spanish Empire 17.5% French Americas 13.6% British North America 6.45% English Americas 3.25% Dutch West Indies 2.0% Danish West Indies 0.3%
Integrated over 10,000 years of history there were probably far more slaves in Africa from Africa, but the numbers are harder to come by.
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Various parts of the USA (i.e. the South) were very late in abolishing slavery, but some parts of the USA abolished slavery around the same time as the UK (and other countries).
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Various Middle Eastern countries (e.g. Omani Empire which had the eastern coast of Africa) participated in the East African Slave Trade (a.k.a. Arab Slave Trade).