  We derive a formula for the unramified Brauer group of a general class of
rationally connected fourfolds birational to conic bundles over smooth
threefolds. We produce new examples of conic bundles over P^3 where this
formula applies and which have nontrivial unramified Brauer group. The
construction uses the theory of contact surfaces and, at least implicitly,
matrix factorizations and symmetric arithmetic Cohen--Macaulay sheaves, as well
as the geometry of special arrangements of rational curves in P^2. We also
prove the existence of universally CH_0-trivial resolutions for the general
class of conic bundle fourfolds we consider. Using the degeneration method, we
thus produce new families of rationally connected fourfolds whose very general
member is not stably rational.
