  We study in detail the ADHM construction of U(N) instantons on noncommutative
Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. We point
out that the completeness condition in the ADHM construction could be
invalidated in certain circumstances. When this happens, regular instanton
configuration may not exist even if the ADHM constraints are satisfied. Some of
the existing solutions in the literature indeed violate the completeness
condition and hence are not correct. We present alternative solutions for these
cases. In particular, we show for the first time how to construct explicitly
regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. We also
give a simple general argument based on the Corrigan's identity that the
topological charge of noncommutative regular instantons is always an integer.
