So far the studies that have been performed in orbit have shown that plants grow perfectly normal (shoots up, roots down--so to speak) in microgravity. They also produce healthy offspring which can grow new plants in orbit.
Experiments have been performed on Arabidopsis and other Brassicaceae.
Aboard the ISS a special “plant growth chamber” called Advanced Astroculture (or ADVASC) is used for Arabidopsis. Within this enclosed chamber the environmental and growth conditions of the plants can be controlled.
The ADVANCED ASTROCULTURE™ controls temperature, humidity, light, atmospheric conditions and delivery of nutrients to plants. It requires no power during Shuttle ascent and descent.
The seeds are planted in a tray containing a special material composed of crushed clay (Arcillite) for rooting. The plant receives its nutrition via basins in ADVASC that contain water and other nutrients.
Another unit with plant growth chambers called the Advanced Biological Research System (ABRS) has also been used on the ISS for Arabidopsis.
Vegetables have been grown in space as well. Mizuna, a type of Japanese lettuce, has been grown in the Lada Validating Vegetable Production Unit. Experiments have been performed in the Lada unit since 2002.
This year (2013) the ISS will begin to grow its first crop of lettuce that will actually be eaten on board! Red-leaved romaine lettuce will be grown under LED lighting starting in December (2013) with the goal for it to eventually be eaten by the astronauts. The plants are expected to grow to edible size in 28 days.