  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) 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
is a separable C*-algebra whose irreducible representations all have dimension
at most N (for some finite N), and if for each n the space of n-dimensional
irreducible representations has finite covering dimension, then A is a
recursive subhomogeneous algebra. We demonstrate the good properties of this
class by proving subprojection and cancellation theorems in it.
