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The Jones-Witten theory gives rise to representations of the (extended)mapping class group of any closed surface Y indexed by a semi-simple Lie groupG and a level k. In the case G=SU(2) these representations (denoted V_A(Y))have a particularly simple description in terms of the Kauffman skein moduleswith parameter A a primitive 4r-th root of unity (r=k+2). In each of theserepresentations (as well as the general G case), Dehn twists act astransformations of finite order, so none represents the mapping class groupM(Y) faithfully. However, taken together, the quantum SU(2) representations arefaithful on non-central elements of M(Y). (Note that M(Y) has non-trivialcenter only if Y is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2punctures, or the closed surface of genus = 2.) Specifically, for a non-centralh in M(Y) there is an r_0(h) such that if r>= r_0(h) and A is a primitive 4r-throot of unity then h acts projectively nontrivially on V_A(Y). Jones' [J]original representation rho_n of the braid groups B_n, sometimes called thegeneric q-analog-SU(2)-representation, is not known to be faithful. However, weshow that any braid h not= id in B_n admits a cabling c = c_1,...,c_n so thatrho_N (c(h)) not= id, N=c_1 + ... + c_n.

QuantumSU(2) faithfully detects mapping class groups modulo center