My lot (3 of mine and their 3 cousins, ages 5-10) love Carcassonne (a tile board game where you "build" a board map with castles and roads and put little men on them to score points. Very visual and no reading required once you know the rules.
Fluxx is great fun (once they can read: best for age 6+): several theme versions of this card game. Our favourite version is pirate fluxx (me hearties!)
Another (card) game is Kinder Bunnies (up to 5 players) : nicking carrots off the other bunnies and playing cards/dice rolling. Comes with two decls in the pack, sky blue is reading-free and when they get older add in the yellow cards.
Robo Rally is great (up to six players): board game where you move your little robot over a maze of conveyor belts, they all bump into each other, fall down pits, get squashed etc. bit more complicated so probably 7yrs +.
qwirkle Cubes is good and can be played with tons of players... No reading needed, just colour matching.
Ooh and tantrix! There are several varients but the standard game is up to 4 players. No reading needed once rules known.
Oops, appear to be a game addict