  We study the cylinder-plate and the cylinder-cylinder Casimir interaction in
the $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of
massless scalar fields. Different combinations of Dirichlet (D) and Neumann (N)
boundary conditions are imposed on the two interacting objects. For the
cylinder-cylinder interaction, we consider the case where one cylinder is
inside the other, and the case where the two cylinders are outside each other.
By computing the transition matrices of the objects and the translation
matrices that relate different coordinate systems, the explicit formulas for
the Casimir interaction energies are derived. Using perturbation technique, we
compute the small separation asymptotic expansions of the Casimir interaction
energies up to the next-to-leading order terms. The leading terms coincide with
the respective results obtained using proximity force approximation, which is
of order $d^{-D+1/2}$, where $d$ is the distance between the two objects. The
results on the next-to-leading order terms are more interesting and important.
We find some universal behaviors. It is also noticed that for the case of
Dirichlet-Dirichlet cylinder-plate interaction, the next-to-leading order term
agrees with that obtained using derivative expansion. Hence, based on our
results on other boundary conditions and on the cylinder-cylinder interaction,
we postulate a formula for the derivative expansion to expand the Casimir
interaction energy up to the next-to-leading order terms for DD, DN, ND and NN
boundary conditions, for the interaction between two curved surfaces in
$(D+1)$-dimensional Minkowski spacetime. It is found that the postulate agrees
with our previous results on the sphere-sphere interactions except when $D=4$.
