We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form
C
(
X
,
M
n
)
C (X, M_n)
many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies, in particular, that if
A
A
is a separable C*-algebra whose irreducible representations all have dimension at most
N
>
∞
,
N > \infty ,
and if for each
n
n
the space of
n
n
-dimensional irreducible representations has finite covering dimension, then
A
A
is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.