  Slice regular functions have been extensively studied over the past decade,
but much less is known about their boundary behavior. In this paper, we
initiate the study of Julia theory for slice regular functions. More
specifically, we establish the quaternionic versions of the Julia lemma, the
Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, and the
Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit
ball $\mathbb B$ and of the right half-space $\mathbb H^+$. Our quaternionic
boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity
of quaternions. Together with some explicit examples, it shows that the slice
derivative of a slice regular self-mapping of $\mathbb B$ at a boundary fixed
point is not necessarily a positive real number, in contrast to that in the
complex case, meaning that its commonly believed version turns out to be
totally wrong.
