It might seem very surprising at first. However, the truth is that, yes, you can certainly grow plants without soil. It was actually discovered hundreds of years ago that the prime purpose of the soil is to hold the minerals for the plants. What it does is that it provides the plant seed the perfect environment to grow in much like what the human womb provides to a fetus.
Given the same environment in another external media such as water, a plant can most certainly grow. This branch of botany is called hydroponics and more efficient methods are still being researched.