As there may have been hundreds of American artists living in Europe during the late 19th century, I will discuss only three: Whistler, Cassatt, and Sargent.

James McNeill Whistler was born in Lowell, Massachusetts in 1834 and lived with his family in Russia. He enrolled in West Point Academy and moved to Paris in 1855 to become an artist.
In 1859 he moved again, this time to London (where he would die in 1903).

Whistler’s most famous painting is of his mother, "Arrangement in Grey and Black No. 1." Painted in 1871, it now resides in the Musée D’Orsay, Paris.
Whistler used "symphonies, variations, nocturnes, arrangements, and harmonies" in the titles of his paintings. He found a connection between: music, painting, and writing.

Mary Cassatt was born in Alleghany City, Pennsylvania in 1844, studying at the Pennsylvania Academy of Fine Arts. Once returning to Europe (and living for the remainder of her life), she copied Old Master paintings, and studied with Jean-Léon Gérôme (as did fellow Pennsylvania artist Thomas Eakins).

Cassatt traveled to Pennsylvania in 1870, but again returned to Paris in 1873.
In 1877, Edgar Degas invited Cassatt to join the Impressionists; she would become the only American artist included. She exhibited in only two of the eight Impressionists’ exhibitions, but she would remain in the group for fourteen years.

Cassatt returned to the US only three other times; she died in 1926 near Paris and was buried in her family plot in France. She allegedly called herself "an American citizen" in her will.

John Singer Sargent was born in 1856 in Italy to expatriate parents (his father was a Philadelphia eye surgeon). He studied art at the age of twelve in Rome and Florence and studied with Carolus-Duran at age eighteen. In 1877 he entered his first major portrait (of his friend Fanny Watts) to the Salon. In 1886 he moved from Paris to London (until his death in 1925).

His family portraits, most notably "The Daughters of Edward Darley Boit" (1882), were originally not intended for museums. This famous painting now resides in the Museum of Fine Arts, Boston.

You can own a framed art print of John Singer Sargent's "The Daughters of Edward Darley Boit."