Picasso, Van Gogh, Matisse, Toulouse Lautrec—and Pointillism, fauvism, Cubism and other early 20C movements
Many of the experimental French artists of the late 19th century were not actually technically Impressionists—though many were their friends and moved in and out of each other's circles. Their smaller movements or individual styles are usually lumped together as "Post-Impressionist."
Artists & examples of Post-Impressionist art in France
Cézanne (1839-1906) may have adopted the short brushstrokes (which he eventually developed into a mosaic of colored rectangles), love of landscape, and light color palate of his Impressionist friends, but he was more formal and deliberate about it, studying his predecessors and seeking to give his modern art a monumentality and permanence, even if the subjects were simple (still lifes, portraits, and focused landscapes).
Gauguin (1848-1903) could never settle himself or his work, trying Brittany first, where he developed sythetism (black outlines around solid colors), and later hopping around the South Pacific, where he was inspired by local styles and colors.
Seurat (1859-91), Signac (1863-1935), and Pissaro together developed divisionism and its more formal cousin pointillism—rather than mixing, say, yellow and blue paint together to make green, they applied tiny dots of yellow and blue right next to each other so that the viewer's eye mixes them together to make green (TV sets and billboards use the same technique).
Toulouse-Lautrec (1864-1901) is most famous for his work with thinned down oils to create paintings and posters of wispy, fluid lines anticipating Art Nouveau and often depicting the Bohemian life of Paris (dance halls, cafes, top-hatted patrons of fancy parties).
Van Gogh (1853-90) was Dutch but spent most of his tortured artistic career in France. He combined divisionism, synthetism, and a touch of Japanese influence, and painted with thick, short strokes. He was never particularly accepted by any artistic circle, and the most popular painter in the world today (his paintings auction for record sums, and he sells more postcards and posters than any other artist) sold only one painting in his short life.
Matisse (1869-1954) and friends took a hint from sythetism and added wild colors and strong patterns to create Fauvism ("wild beasts"). Matisse continued exploring these themes even when most were turning to Cubism, and later in life (with failing health) began assembling brightly colored collages of paper cutouts.
Braque (1882-1963) and Spanish-born Picasso (1881-1973) decided to try and paint objects from all points of view at once, accepting that the canvas is flat and merely representing all visible angles by painting them adjacent to each other rather than using optical tricks like perspective to fool viewers into seeing three dimensions. The fractured, imploding-looking result was called Cubism, and was expanded upon by the likes of Léger (1881-1955) and the Spaniard Gris (1887-1927), while Braque started to collage in bits of paper and cardboard, and Picasso moved on to other styles (you can see all of his periods at museums dedicated to him in Paris, Antibes, and Vallauris, where Picasso revived the ceramics industry).
Chagall (1889-1985) is hard to pin down, a Hasidic Jew who traveled widely in Europe, the U.S., Mexico, and Israel. His art started from Cubism and picked up inspiration everywhere to fuel an idiosyncratic, brightly colored, allegorical, often whimsical style. There's a museum devoted to him in Vence and another in Nice, plus several stained glass windows in the Reims cathedral and the ceiling of Paris's Opera Garnier.