Japanese cars came to the American market slowly, first with underpowered rustbuckets and imitations of American and European vehicles, then with underpowered, efficient, and reliable vehicles. As Japanese cars moved upscale, though, a gap opened at the bottom, and Korean automakers happily filled it, starting with the Pontiac LeMans (Daewoo), Ford Aspire (Kia), and Hyundai Excel. Now, Hyundai and Kia are joined and offer similar vehicles with the best warranties in the business, while Daewoo is owned by General Motors.
The Korean manufacturers, while they don't have a massive market share - yet - have come a long way. Their gains in technology, quality, and "feel" are astounding. Hyundai in particular now beats a number of Japanese and European brands in quality ratings.