The one-dimensional isotropic quantum Heisenberg spin systems with ran
dom couplings and random spin sizes are investigated using a real-spac
e renormalization-group scheme. It is demonstrated that these systems
belong to a universality class of disordered spin systems, characteriz
ed by weakly coupled large effective spins. In this large-spin phase t
he uniform magnetic susceptibility diverges as T-1 with a nonuniversal
Curie constant at low temperatures T, while the specific heat vanishe
s as T-/alpha//lnT/ for T-->0. For a broad range of initial distributi
ons of couplings and spin sizes the distribution functions approach a
single fixed-point form, where alpha(approximate to) -0.44. For some s
ingular initial distributions, however, the fixed-point form of distri
butions becomes nonuniversal, suggesting that there is a line of fixed
points.