UBC Garden, located in Vancouver, British Columbia, is Canada's oldest university botanical garden. It was founded in 1916. Today it covers over 100 acres, and contains about 12,000 plants from temperate regions all over the world.
There are several gardens within the garden. These include the Alpine Garden, Asian Garden, Carolinian Forest, Food Garden, and Native garden. Visitors will see magnolias, rhododendrons, dogwood, roses, and a plethora of Alpine flora.
The garden is open daily from 9:30 AM - 5:00 PM. The best times to visit are April and May, or September and October. Cost of admission is $8.00 for adults and $4.00 for children.
Zagreb Botanical Garden, located in Zagreb, Croatia, was built in 1889. It was originally designed to be a research area for the Zagreb University botany department.
The garden covers about twelve acres and contains 10,000 different species of plants. Visitors can see an arboretum, 14 greenhouses, 3 rock gardens, 2 lakes, and an ornamental bridge.
The garden is open from April 1 to November 1. Hours are Monday and Tuesday from 9:00 AM to 2:30 PM. Wednesday through Sunday from 9:00 AM to 6:00 PM. Admission is free.
Yuan Ming Yuan, translated, "Garden of Gardens," is located in Dunhuang, China. It was built in 1707 by the Kangxi emperor, and at one time was considered China's largest and most luxurious imperial garden. 
Today most of it lies in ruins. Much of it was destroyed by French and British troops in the Second Opium War of 1860. Still, it's an interesting place to visit. On its 865 acres, visitors will see ruins of ancient imperial buildings, along with hills, lakes, ponds, woods, and flowers.
Some restorations have begun on the garden. Wanhua (10,000 Flower Maze), a maze of winding paths between low walls, was recently reconstructed in the Changchun Yuan (Garden of Eternal Spring) section.
Located in Xishuangbanna, China, the Xishuangbanna Tropical Botanical Garden covers an area of almost 2,000 acres. It is China's largest tropical garden. The full name of the garden is Tropical Plant Research Institute of the Chinese Academy of Science. (Big name for a big garden!) As you can guess, the garden is used for research, and contains over 10,000 species of tropical plants.
Wuhan Botanical Garden, located in Hubei, China, covers over 150 acres and contains over 4,000 species of flora. It was formed in 1956 as a place for scientists to study plants native to China.
The garden is open daily from 8:00 AM - 5:00 PM. Admission is 30 yuan.
Villa Lante Garden, located in Bagnaia, Italy, is a formal Italian garden that covers about fifty acres. It was designed for Cardinal Gambara in 1568 who lived there in the Villa. It's a tribute to the lavish lifestyle for those high ranking officials in the Catholic Church at that time.
As you wander through the garden, you will sees statues, grottos, box hedges, and of course, a lot of fountains.
The garden is open everyday except Monday. Hours are 9:00 AM - 4:30 PM in the winter, and 9:00 AM - 7:30 PM in the summer. Admission is 2 euros.
Tibilisi Botanical Garden is located in Tibilisi, Georgia. (That's Georgia near Russia - not the United States.) It covers over 325 acres and contains roughly 3500 different types of plants.
Shenyang Botanical Garden, located in Shenyang City, China, covers 467 acres and contains more than 1,000 species of flowers, trees, and shrubs. 
As you walk through the garden, you'll see several ponds, waterfalls, and ornate bridges. There are 20 gardens in all. These include an azalea garden, peony garden, lilac garden, and rose garden.
Rikugien Garden, located in Tokyo, Japan, is considered one of the area's most beautiful gardens. It was built between 1695-1702 by Yanagisawa Yoshiyasu, and in 1938 was donated to the Tokyo city government.
The garden is open daily from 8:30 AM to 4:30 PM.