In classical French cuisine, a brown sauce generally refers to a sauce with a meat stock base, thickened by reduction and sometimes the addition of a browned roux, similar in some ways to but more involved than a gravy. The classic mother sauce example is espagnole sauce as well as its derivative demi-glace, though other varieties exist.
Swedish meatballs in brown sauce, as served at IKEA
In Danish cuisine brown sauce (brun sovs) is a very common sauce, and refers to a sauce with a meat stock base (in modern times, often replaced by broth made from bouillon cubes), thickened by a thickening starch agent, such as flour or cornstarch, and colored a rich, deep brown with a product consisting of dark caramelized sugar, known as brun kulør (literally, "brown colouring") or madkulør (literally, "food colouring"). It is similar to what is known in the U.S.A. as a brown gravy. Variations include mushroom sauce, onion sauce, and herbed brown sauce.
The Norwegian variety (brun saus) is made in the same way as the Danish brown sauce, usually from wheat flour. The sauce is colored by browning the wheat flour in the pan or adding food coloring (sukkerkulør, literally "sugar coloring"). The sauce may acquire different tastes depending on the meat served, as it's common to cook the meat for a while in the sauce before serving.
In Sweden (brunsås) and Finland (ruskeakastike), meatballs are usually served with a brown, thick sauce, prepared in the same manner as espagnole sauce (combining dark brown roux with stock).