  Quantum magnets on kagome lattice geometries in two and three spatial
dimensions are archetypal examples of spin systems in which geometric
frustration inhibits conventional magnetic ordering and instead benefits the
emergence of long-range entangled spin liquids at low temperature. Here we
employ a recently developed pseudo-fermion functional renormalization group
(pf-FRG) approach to study the low-temperature quantum magnetism of kagome and
hyperkagome spin systems with exchange interactions beyond the nearest neighbor
coupling. We find that next-nearest neighbor couplings stabilize a variety of
magnetic orders as well as induce additional spin liquid regimes giving rise to
rather rich phase diagrams, which we characterize in detail. On a technical
level, we find that the pf-FRG approach is in excellent quantitative agreement
with high-temperature series expansions over their range of validity and it
exhibits a systematic finite-size convergence in the temperature regime below.
We discuss notable advantages and some current limitations of the pf-FRG
approach in the ongoing search for unconventional forms of quantum magnetism.
