How many solar systems are there in the universe?
That is a very good question but we don't quite know the answer to it yet.
There are a number of projects running at the moment that hope to try and provide some answers, such as NASA's Kepler mission and the European CoRoT, as well as various ground-based efforts. So far they are suggesting that a reasonable fraction of stars (around 10 per cent or even more) have planets, but so far we have not really able to find systems quite like our own.
In terms of the number of solar systems present in the universe, there are something like 300 billion stars in the Milky Way, so if 10 per cent of them have planets there are around 30 billion planets in our galaxy alone, and there are over 100 billion galaxies in the observable Universe for a total of something in the order of 10^21 (that's 1 then 21 zeros) planets in the observable Universe. There is still quite a bit of uncertainty in that number however, and we don't yet know how many of them would look like our solar system.
Hope this helps! :D