For a large class of self-similar random sets
F
F
in
R
d
\mathbb {R}^d
, geometric parameters
C
k
(
F
)
C_k(F)
,
k
=
0
,
…
,
d
k=0,\ldots ,d
, are introduced. They arise as a.s. (average or essential) limits of the volume
C
d
(
F
(
ε
)
)
C_d(F(\varepsilon ))
, the surface area
C
d
−
1
(
F
(
ε
)
)
C_{d-1}(F(\varepsilon ))
and the integrals of general mean curvatures over the unit normal bundles
C
k
(
F
(
ε
)
)
C_k(F(\varepsilon ))
of the parallel sets
F
(
ε
)
F(\varepsilon )
of distance
ε
\varepsilon
rescaled by
ε
D
−
k
\varepsilon ^{D-k}
as
ε
→
0
\varepsilon \rightarrow 0
. Here
D
D
equals the a.s. Hausdorff dimension of
F
F
. The corresponding results for the expectations are also proved.