In many cases, there aren't any huge issues if you're hit by an uninsured driver. You may be able to seek compensation directly, of course, but your own insurance may help cover the costs if the other driver doesn't have the proper policy.
One important thing to note, though, is that the coverage you get from your insurance company is typically for compensatory damages. These are very tangible costs that are linked directly to the crash.
For example, perhaps you broke both of your legs in the accident. You needed transportation to the hospital, which was expensive. You spent multiple nights in the hospital after the initial treatment. This meant that you had to pay for the treatment and possibly lose wages since you couldn't go to work. After getting out, you may have missed even more time at work since it was impossible to do your job until you healed.
These are all costs that may be covered. However, the entire experience may have been stressful and emotionally trying. You may have endured a lot of pain and suffering. While you may be able to start a civil lawsuit to seek compensation for these types of things, your auto insurance generally isn't going to cover you on these fronts.
As you can see, it's very important to know exactly what types of compensation to expect and where to get them. After a car accident, be sure you look into all of your options. Even if you are getting payments from your insurance company, it may not be wise to assume that these are all you can get. You just have to know what legal steps to take to seek out the rest.
Source: 360 Degres of Financial Literacy, "Uninsured/Underinsured Motorist Coverage," accessed Dec. 30, 2016