Kiwiberries are quite resistant to disease, but they are susceptible to phytophthora root rot in excessively moist soil.  Conversely, soil that dries quickly can dramatically slow growth – for at least a few years.  Although the complaint seems to be quite rare, the spotted wing drosophila (Asian fruit fly) may be a problem in regions that have a longer grow season than those typically found in the north, otherwise they don't have any significant pests either.  Hardy kiwi (Actinidia arguta) occasionally experience one relatively serious issue, though.  Productivity may begin to fade at some point, or fail to produce well from the start, but a heavy pruning can solve this problem. 

Kiwiberries perform well if they are grown fairly close to the ocean or a great lake.  Most other places have a very unstable climate, so their fruiting shoots often suffer from frost damage.  However, damage strictly to young leaves can be insignificant.  Some varieties developed in the south may awaken much later than most, but they are quite new and none of them are from species referred to as kiwiberries. 

Hardy kiwi are borderline invasive and may take 4-7 years to produce – with the primary exception being Issai, which can fruit very quickly.  Arctic kiwi are easier to control and begin to produce much sooner.

Hardy Kiwi: Actinidia arguta

Anna

🔍
Originated in Russia
Alias
• Ananasnaya, Annasnaya
Zone
• 4b
Ripens
🥝
• late
Color
• green skin with a purple-red blush
Fruit
• small size
Yield
• highly productive

Out of the 6+ varieties I have seen, Anna develops the most attractive fall colors in my warm-summer humid continental (Dfb) climate.

Chang Bai

🔍
Seedling of Qui.  Selected in China near the border of North Korea.
Zone
• 4b
Color
• green skin
Fruit
• medium-small size
• flat pumpkin shape

I have not heard of any reports from other people, but in my experience, the quality of Chang Bai has been quite variable.  However, the vine has a different aesthetic than the other hardy kiwis I have seen, and I think it is quite nice.

Cherry Bomb

🔍
Zone
• 4
Color
• red skin

Cherry Bomb might be a cross between A. arguta and A. kolomitka.  If so, it may be able to tolerate a hardiness zone lower than what is generally considered to be the limit for most available hardy kiwi varieties (4b).

Cordifolia

🔍
Zone
• 4b
• low chill
Color
• dark green skin
Fruit
• relatively sweet?

Cordifolia may also be referred to as 'Meyer's Cordifolia' to avoid confusion with another variety that is now being referred to as 'Early Cordifolia', but I don't know the full story.

Dumbarton Oaks

🔍
Selected in Georgetown, Washington D.C.
Zone
• 4b
Ripens
🥝
• early-mid
• ~3 to 4 weeks before Anna
Color
• light green skin
Fruit
• pumpkin-like shape

Fifty Five

🔍
Zone
• 4b
Ripens
🥝
• early
• roughly 6 weeks before Anna

Flowercloud Male

🔍
A. arguta x A. kolomikta
Zone
• 4b?
Ripens
🥝
•    ◦ precocious

Flowercloud may begin to flower as soon as the year after planting, but it "might" bloom too early in "some" climates due to its hybrid parentage.

Fortyniner

🔍
Developed in Chico, California.
Zone
• 4b
Ripens
🥝
• mid-late
• somewhere between Dumbarton Oaks and Anna

Fortyniner might be the same variety as 74-49.

Geneva

🔍
Selected in Geneva, New York.
Zone
• 4b
Ripens
🥝
• mid-late?
• somewhere between Dumbarton Oaks and Anna

Hardy Red

🔍
Zone
• 4b
Color
• red skin

Hardy Red has supposedly done well in taste tests, according to nurseries.  I have not seen any results or heard about this variety elsewhere, though.

Issai

🔍
Originated in Japan
Zone
• 4b
• 300 chill hours
Flowers
• partially self-fertile (parthenocarpy)
Ripens
🥝
•    ◦ precocious
Color
• green skin

Issai grows less vigorously than most arguta varieties.

Jumbo

🔍
Originated in Italy
Zone
• 4b
Color
• green-yellow skin
Fruit
• medium-small size

Jumbo produces large berries for a hardy kiwi, roughly twice the size of Anna.  It may be the same variety as MSU (Michigan State).

Ken's Red

🔍
A. arguta x A. melanandra.  Originated in New Zealand.
Zone
• 4b? (5)
• low chill
Ripens
🥝
• late?
• somewhat biennial?
Color
• red skin

Meader Male

🔍
Zone
• 4b (4a)    ◦ one report claims it survived down to -28F (-33.3C)

Natasha

🔍
Originated in Vladivostok, Russia.
Zone
• 3b

Ogden Point

🔍
Zone
• 4b

Prolific

🔍
Zone
• 4b
Flowers
• partially self-fertile (parthenocarpy)
Color
• dark green skin

Red Princess

🔍
Cordifolia (A. arguta) x A. melanandra.  Originated in New Zealand.
Zone
• 4b? (5)
Color
• red skin

Rossana

🔍
Selected in Italy
Zone
• 3b

Tatyana

🔍
Zone
• 3b
Color
• light green skin

Arctic Kiwi: Actinidia kolomikta

Arctic Beauty Male

🔍
Zone
• 3a

The leaves of Arctic Beauty develop white and pink colors, especially when they are exposed to the sun.  It may last for a month or so, but it then fades out into a light tan color.

Emerald

🔍
Zone
• 3a
Color
• light green skin

Frost

🔍
Zone
• 3a
Ripens
🥝
• early? (early-mid)

Frost ripens earlier than most Kolomikta varieties.

Hero

🔍
Selected from the wild near Vladivostock, Russia (Far East).
Alias
• Dostoynaya
Zone
• 3a
Color
• light green skin

'Hero' has an unconfirmed report about being partially self-fertile.

Pasha Male

🔍
Zone
• 3a

The leaves of Pasha develop white and pink colors.  If 'Pasha Male' is similar to 'Arctic Beauty Male', the colors are amplified by exposure to the sun.

Red Beauty

🔍
Zone
• 3a

September Sun

🔍
Originated in Russia
Alias
• Sentyabraskaya
Zone
• 3a
Ripens
🥝
• mid?

September Sun is the only female I know of that produces ornamental leaves similar to that of the male arctic kiwi.

Viktor

🔍
Selected from the wild north of Vladivostok, Russia.
Zone
• 3a

Viktor produces larger berries than most kolomikta varieties.

Golden Kiwi: Actinidia chinensis

AU Golden Dragon

🔍
Zone
• 7 (6b)
• 800 chill hours
Ripens
🥝
• late?
• matures roughly 7 weeks earlier than AU Fitzgerald
• mid-late August in central Alabama
Color
• yellow-green flesh
Fruit
• large-size, 89 g/fruit

Golden Dragon can hang on the vine for roughly a month after maturing, but the fruit will become very soft.  This may be typical for many chinensis and deliciosa varieties, and they may be able to hang much longer than that, provided it doesn't get too cold.

AU Golden Sunshine

🔍
Selected in Alabama from material that originated in Chongyang County of China (Hubei Province).
Zone
• 7 (6b)
• 700 chill hours
Buds
• break in early April in central Alabama (later than most varieties from any species)
Blooms
💮
• late    ◦ mid-late April in central Alabama
Ripens
🥝
• very late?
• matures mid September in central Alabama
• ~4 weeks before AU Fitzgerald
Color
• yellow flesh
Fruit
• large-size, 90 g/fruit

AU Golden Tiger Male

🔍
AU Golden Dragon x unknown male (pollination occured in China).  Selected in Auburn, Alabama.
Zone
• 7 (6b)
Buds
• break later than most (along with AU Golden Sunshine)
Blooms
💮
• late    ◦ along with AU Golden Sunshine

Fuzzy Kiwi: Actinidia deliciosa

Actinidia deliciosa requires a long, hot grow season to properly ripen the fruit.

AU Authur Male

🔍
Zone
• 8a (7)
• less than 800 chill hours

AU Fitzgerald

🔍
may be a seedling of Hayward.  Developed in Summerdale, Alabama.
Zone
• 8a (7)
• 550-800 chill hours
Ripens
🥝
• extremely late
• matures mid-October in central Alabama
Fruit
• medium-large size, 70 g/fruit

Hayward

🔍
chance seedling.  Discovered in Auckland, New Zealand (1920's).
Zone
• 7    ◦ down to -2F (-18.9C)
• 600-800 chill hours
Ripens
🥝
• extremely late
Storage
• up to 6 months

Hayward is the most commercially cultivated variety.