  We consider tromino tilings of $m\times n$ domino-deficient rectangles, where
$3|(mn-2)$ and $m,n\geq0$, and characterize all cases of domino removal that
admit such tilings, thereby settling the open problem posed by J. M. Ash and S.
Golomb in \cite {marshall}. Based on this characterization, we design a
procedure for constructing such a tiling if one exists. We also consider the
problem of counting such tilings and derive the exact formula for the number of
tilings for $2\times(3t+1)$ rectangles, the exact generating function for
$4\times(3t+2)$ rectangles, where $t\geq0$, and an upper bound on the number of
tromino tilings for $m\times n$ domino-deficient rectangles. We also consider
general 2-deficiency in $n\times4$ rectangles, where $n\geq8$, and characterize
all pairs of squares which do not permit a tromino tiling.
