This is a very important question. Currently there are three main ways in which waste is dealt with.
First and the most common way is to dump waste into landfills. The benefit of this is threefold. First, you can put it a lot of garbage in a landfill. Second, it is cost effective, as you can imagine. Third, if the landfill is created well, then it is effective and safe.
The second way garbage is disposed of is through incineration. This, as you might imagine, is more controversial. However, it is still a popular way, especially when it comes to certain types of wastes, like medical waste and other hazardous things. Also this is a popular method in countries where there is a shortage of land.
The last method is some waste is turned into energy. This sounds great in theory, but there needs to be more work in this area.
As you can see, waste management is an important topic and there are several things we can do to help in this area. The most important and easiest thing to do is consistently to recycle. Another thing we can do is reuse. This produces less waste. Finally, we can buy recycled products.