  We develop the inhomogeneous counterpart to some key aspects of the story of
the Duffin--Schaeffer Conjecture (1941). Specifically, we construct
counterexamples to a number of candidates for a sans-monotonicity version of
Szusz's inhomogeneous (1958) version of Khintchine's Theorem (1924). For
example, given any real sequence {y_i}, we build a divergent series of
non-negative reals psi(n) such that for any y in {y_i}, almost no real number
is inhomogeneously psi-approximable with inhomogeneous parameter y.
Furthermore, given any second sequence {z_i} not intersecting the rational span
of {1,y_i}, and assuming a dynamical version of Erdos' Covering Systems
Conjecture (1950), we can ensure that almost every real number is
inhomogeneously psi-approximable with any inhomogeneous parameter z in {z_i}.
Next, we prove a positive result that is near optimal in view of the
limitations that our counterexamples impose. This leads to a discussion of
natural analogues of the Duffin--Schaeffer Conjecture and Duffin--Schaeffer
Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove
versions of Gallagher's Zero-One Law (1961) for inhomogeneous approximation by
reduced fractions.
