  In the problem of classical group testing one aims to identify a small subset
(of expected size $d$) diseased individuals/defective items in a large
population (of size $n$) based on a minimal number of suitably-designed group
non-adaptive tests on subsets of items, where the test outcome is governed by
an "OR" function, i.e., the test outcome is positive iff the given test
contains at least one defective item. Motivated by physical considerations we
consider a generalized scenario (that includes as special cases multiple other
group-testing-like models in the literature) wherein the test outcome is
governed by an arbitrary $\textit{monotone}$ (stochastic) test function
$f(\cdot)$, with the test outcome being positive with probability $f(x)$, where
$x$ is the number of defectives tested in that pool. For any monotone test
function $f(\cdot)$ we present a non-adaptive generalized group-testing scheme
that identifies all defective items with high probability. Our scheme requires
at most ${\cal O}(d^2\log(n))$ tests for any monotone test function $f(\cdot)$,
and at most ${\cal O}(d\log(n))$ in the physically relevant sub-class of
$\textit{sensitive}$ test functions (and hence is information-theoretically
order-optimal for this sub-class).
