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.
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).
Slavery in Mauritania dates back centuries and continues to this day (so this much more than covers the time when there were slaves in the US).
The French outlawed slavery in 1905 but were unable to effectively enforce the ban. It was abolished again in 1981 by the Mauritanian government but enforcement has been weak to say the least.
Slavery in Mauritania is hardly ‘modern.’ It is an institution deeply rooted in the history of the country and region. The ruling minority Beydanes (Arab-Berbers) historically enslaved Haratin (or “Black Moors”).
Slaves were sold to Europeans from Mauritania during the Atlantic slave trade, but thousands were also enslaved locally:
Since 1677, the Arabs won a battle with the Berbers, putting Arabs in control, with Berbers in the middle (divided into 2 groups: religious scholars called zawiyas and farmers and herders called znaga), and the blacks on the bottom (also divided into 2 groups: former slaves called haratani and slaves, abid).
No one knows for sure how many people are enslaved in Mauritania today but a conservative estimate is 43,000 (1% of the population). In 2014 - more than 30 years after abolition - the number was estimated at 140,000 (3.5% of the population).
Slavery was also widespread in parts of West Africa (e.g. Mali, Ivory Coast, Burkina Faso) up until the late 19th century. Only the defeat of the Malinke ruler Samori Toure in 1898 allowed the French to effectively put an end to the practice.
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).