My son (aged 4) has begun to sleep in his own bed, in his own room. But, almost every night, he wakes up saying he had a bad dream. What do I do to tackle this problem?

As this is still a transition for him, maybe you can try sleeping with him in his room for the time being. However, you need to let him know that this is only temporary. After he seems to be more settled, you can stay in his room till he falls asleep and then return to your own room. Hope it works. :)