While there is no hard and fast rule or dress code as such in Kuwait, it is important to know how to dress when you're in a foreign country, as there are varying degrees of tolerance from around the world to clothing types.

National Dress Code
There is no compulsory Islamic dress code in Kuwait, and generally Kuwaiti men are seen wearing thwab (an ankle-length white cotton shirt), while women are seen wearing abaya (the black over-garment covering most parts of the body).

Kuwaiti men
Kuwaiti men are always found with a thobe (an ankle-length robe that is mostly made of cotton). There are many different types of robes and men can wear them even for official business including meetings, and when attending different occasions. A bisht (outer cloak) can be worn during special occasions.

Kuwaiti women
Traditional Kuwaiti women when running errands are normally seen wearing their traditional costumes (black abayas) with their whole body covered from head to toe. Some Kuwaiti women also prefer to cover their face and hands too. They usually cover their hair too.
Some Kuwaiti women are also sometimes seen in western clothes. Although they may choose from the more demure styles, the latest designs are worn, irrespective of the climate and convenience. However, their traditional clothing such as the thob (the long over-dress), is still used for dancing on festive occasion. When in public, several Kuwait women cover their chic western clothing with an aba (the silky black coat). Bedouin women may also wear a burqa (the short black veil that covers the entire face). Modest clothing is highly valued for women, particularly, as it raises respect for her and her perceived sense of self-worth.

Expat men
There is no official dress code for expat men and women in Kuwait. For Kuwaiti men, fashion is relatively simple. During those colder months they wear darker coloured dishdashas, the rest of the year their dishdashas are white. Non-Kuwaiti men (and less traditional Kuwaitis too) can wear tshirts and even trousers showing part of their legs, but shorts are not recommended: no one wears those and you will most likely get some looks. Three-fourths is as much as is acceptable really.

Expat women
As a woman, if you want to earn respect, you shouldn't show cleavage, or have bare shoulders, or wear skirts or shorts that are above the knee level.
On the whole, it is best to dress in loose light-weight clothing without much skin show. Your skirts and sleeves need to be on the long side. If you should wear shorts, wear capris or long walking shorts.  Loose linen-cotton trousers or decent length skirts and tops with short sleeves are just fine. Tanktops are not recommended either. A normal tshirt that is not too tight and with normal sleeves should be fine. However, you definitely need not cover your hair or your face.

Office / Business wear
In Kuwait, it is also important to know that suits are rarely worn, and is usually worn only if you want to attend a very important meeting or other social events. When in the office, shirts are normally taken as a standard wear, but they have to be long-sleeved, complimented with a tie and long trousers.
Women should stay away from wearing any revealing or tight-fitting clothing to avoid giving in to offence. The skirts should cover the knee and sleeves should cover the elbow and fasten at the neck.
Light-weight business suits are appropriate for men and women. Suits that are tailored are best for men, while women can wear a pantsuit if necessary. If the weather is not too warm, women can wear pantyhose with a skirt business suit

Bottom line:
Although, Kuwait has the reputation of being a conservative place, the rules are not far too different from other Muslim countries. As long as you cover your knees and shoulders, you are fine in Kuwait. Loose fitting outfit is best in Kuwait, and given the hot weather, materials like cotton and linen are more suitable for the summers here. If you are seen wearing tight dresses, or mini-skirts or tank tops, there are little chances that anything really happens, but, keep in mind this is not your home country, so a bit of respect for local customs would always be appreciated.