Jut what is exposition? The purpose is to inform the reader of things they need to know to understand the characters and the story. This is dealing with information and there are a number of ways to get the information to the reader.
A writer can use a big block of information otherwise known as an info dump. Beward putting too much in one chunk or you will end up sounding like a teacher.
A writer can use one character's observation of another to give bits of information. One problem with this approach is mind reading. Statements are made as to one character's view of a situation in ways that the observing character can't know. Such as Mary was sad or Mary seemed sad or I figured Mary was sad. What the writer has to do is show how this conclusion was drawn. Tears glistened in Mary's eyes. The news of her father's death definitely had an effect.
Dialogue is a good way to give information, but beware having two characters discuss something they already know. This can be boring and make the reader toss the book. There is a way around this and that could be by having the information the writer wants to show become a bit of contention between the pair. Arguments are usually nice places to pop in a bit of information.
Another way is to have the character or characters reminiscence about an event in the past. Take care here not to have this become a single person trip into the past. A lengthy one can pull the reader out of the story. A short sentence can remind the reader of something they had read before, or in the case of a series, of another book. For long memories of the past have two people who were there talk about the event.
When giving information, ash yourself these questions. Is the information absolutely necessary for the reader so he understands the characters and the plot? How does the viewpoint character know this information?