With its traffic, ever-present crowds, and abundance of concrete, Seoul [1] may seem ill-suited to young children initially, but there are plenty of destinations and pockets of the city that will keep them enthralled and give them the opportunity to run free.
Many parents come to appreciate the child-centeredness of Korean society—kids are basically given a lot of space to be themselves, and no one is especially put out by crying (or cavorting) young ones, for example. Children are also entitled to heavily discounted or free rides on public transit and entry to most attractions.
Here are a few places and attractions kids should enjoy.
Links:
[1] http://www.moon.com/destinations/seoul/discover-seoul
[2] http://www.moon.com/destinations/seoul/sights/gyeongbokgung-and-gwanghwamun
[3] http://www.moon.com/destinations/seoul/sights/n-seoul-tower
[4] http://www.moon.com/destinations/seoul/arts-and-leisure/seoul-forest
[5] http://www.moon.com/destinations/seoul/arts-and-leisure/children-s-grand-park
[6] http://www.moon.com/destinations/seoul/sights/national-museum-korea
[7] http://www.moon.com/destinations/seoul/sights/war-memorial-korea
[8] http://www.moon.com/destinations/seoul/sights/lotte-world
[9] http://www.moon.com/destinations/seoul/excursions-seoul/yongin