An integral self-affine tile is the solution of a set equation
A
T
=
⋃
d
∈
D
(
T
+
d
)
\mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d)
, where
A
\mathbf {A}
is an
n
×
n
n \times n
integer matrix and
D
\mathcal {D}
is a finite subset of
Z
n
\mathbb {Z}^n
. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices
A
∈
Q
n
×
n
\mathbf {A} \in \mathbb {Q}^{n \times n}
. We define rational self-affine tiles as compact subsets of the open subring
R
n
×
∏
p
K
p
\mathbb {R}^n\times \prod _\mathfrak {p} K_\mathfrak {p}
of the adèle ring
A
K
\mathbb {A}_K
, where the factors of the (finite) product are certain
p
\mathfrak {p}
-adic completions of a number field
K
K
that is defined in terms of the characteristic polynomial of
A
\mathbf {A}
. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles.
We also associate a second kind of tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with
R
n
×
∏
p
{
0
}
≃
R
n
\mathbb {R}^n \times \prod _\mathfrak {p} \{0\} \simeq \mathbb {R}^n
. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of the digit set
D
\mathcal {D}
, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems.