The first time we hear anything about the Radleys is in chapter 1 of Harper Lee's To Kill a Mockingbird, and Scout tells us that three doors to the south of her home is the Radley Place, which
was inhabited by an unknown entity the mere description of whom was enough to make us behave for days on end....
Soon we hear about Boo Radley, the "malevolent phantom" who lived in the Radley Place. The rumors and gossip surrounding Boo range from claims that he is a peeping tom to a small-time thief; he freezes flowers with his breath, and anyone who ate pecans from one of the Radley pecan trees would certainly be killed.
Despite all these wild rumors, Scout admits that neither she nor Jem has ever even seen their mysterious neighbor to the south.
At the end of chapter 1, Dill dares Jem to run up and touch the Radley's house; after he does it and runs back out of the yard, they see the tiniest flicker of movement near the shutter from inside the house. This is our first possible indication of Boo Radley.
In chapters 2 and 3, Scout is busy trying to navigate her way through the complications of going to school and figuring out how things work. She and Jem pass the Radley Place many times in a day, but there is no sign of Boo.
In chapter 4, Boo begins to leave tiny treats and surprises for the children in the knot of one of an oak tree on the Radley property. First Scout sees the glimmer of two shiny, foil-wrapped sticks of Wrigley’s Double-Mint chewing gum. The next treasure the kids spy is a ring-box covered in bits of tinfoil. Inside are two shiny Indian-head pennies.
Jem rolls Scout inside a tire; he rolls her particularly hard because he's mad at her, and she lands with a plop inside the Radley's front gate. Later she reveals that when she fell out of the tire she heard a low chuckling from inside the Radley house.
In chapter 5, the boys decide to give Boo a note asking him to come out, but no real contact is made. In chapter 6, however, the kids get quite bold. One night they crawl under the fence into the Radleys' back yard and climb on one another to try to get a glimpse of Boo through the window. Jem sees someone, but it is probably not Boo, as the next thing we know Nathan Radley is in the garden with his shotgun. When they run away, Jem is forced to leave his overalls behind because they got caught in a fence.
The most significant interactions the Finch children have with Boo happen in chapters 7 and 8. First, Jem admits that when he went back to retrieve his pants, someone had sewn then (rather crudely, as a boy might do) and left them neatly for him. Second, the kids find a ball of twine in the knot-hole and then they discover
two small images carved in soap. One was the figure of a boy, the other wore a crude dress.
They are legitimate likenesses of the Finch children, presumably carved by Boo. They also find a pocket watch on a chain with a pocket knife, a pack of chewing gum and one old spelling bee medal, but eventually the knot hole gets filled with cement.
The closest either of them gets to Arthur Radley is Scout on the night of Miss Maudie's fire. Scout was shivering in the cold, and in the confusion of the event she did not notice that Boo placed a blanket over her shoulders to keep her warm.
None of these events is particularly significant, but they do add up to a lot of contact between the children and Boo. Despite the fact that they never meet in person, Boo goes to a lot of trouble to make himself known to the Finch children.