  We investigate the spin $S=1/2$ Heisenberg model on the body centered cubic
lattice in the presence of ferromagnetic and antiferromagnetic nearest-neighbor
$J_{1}$, second-neighbor $J_{2}$, and third-neighbor $J_{3}$ exchange
interactions. The classical ground state phase diagram obtained by a
Luttinger-Tisza analysis is shown to host six different (noncollinear)
helimagnetic orders in addition to ferromagnetic, N\'eel, stripe and planar
antiferromagnetic orders. Employing the pseudofermion functional
renormalization group (PFFRG) method for quantum spins ($S=1/2$) we find an
extended nonmagnetic region, and significant shifts to the classical phase
boundaries and helimagnetic pitch vectors caused by quantum fluctuations while
no new long-range dipolar magnetic orders are stabilized. The nonmagnetic phase
is found to disappear for $S=1$. We calculate the magnetic ordering
temperatures from PFFRG and quantum Monte Carlo methods, and make comparisons
to available data
