We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function
B
(
r
,
s
)
B(r,s)
of the natural number parameters
r
r
and
s
s
so that for any system of algebraic ordinary differential-difference equations in the variables
x
=
x
1
,
…
,
x
q
\mathbfit {x} = x_1, \ldots , x_q
and
y
=
y
1
,
…
,
y
r
\mathbfit {y} = y_1, \ldots , y_r
, each of which has order and degree in
y
\mathbfit {y}
bounded by
s
s
over a differential-difference field, there is a nontrivial consequence of this system involving just the
x
\mathbfit {x}
variables if and only if such a consequence may be constructed algebraically by applying no more than
B
(
r
,
s
)
B(r,s)
iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over
C
\mathbb {C}
is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.