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For a fixed prime p, the p-class tower F(p,infinity,K) of a number field K isconsidered to be known if a pro-p presentation of the Galois group H = Gal(F(p,infinity,K)/K ) is given. In the last few years, it turned out that theArtin pattern AP(K) = (tau(K),kappa(K)) consisting of targets tau(K) =(Cl(p,L)) and kernels kappa(K) = (ker(J(L/K)) of class extensions J(L/K):Cl(p,K) --> Cl(p,L) to unramified abelian subfields L/K of the Hilbert p-classfield F(p,1,K) only suffices for determining the two-stage approximation G =H/H" of H. Additional techniques had to be developed for identifying the groupH itself: searching strategies in descendant trees of finite p-groups, iteratedand multilayered IPADs of second order, and the cohomological concept ofShafarevich covers involving relation ranks. This enabled the discovery ofthree-stage towers of p-class fields over quadratic base fields K = Q(squareroot(d) ) for p = 2,3,5. These non-metabelian towers reveal the newphenomenon of various tree topologies expressing the mutual location of thegroups H and G.

Recent progress in determining p-class field towers