  We study existence, uniqueness and asymptotic spatial behavior of
time-periodic strong solutions to the Navier-Stokes equations in the exterior
of a rigid body, $\mathscr B$, moving by time-periodic motion of given period
$T$, when the data are sufficiently regular and small. Our contribution
improves all previous ones in several directions. For example, we allow both
translational, $\bfxi$, and angular, $\bfomega$, velocities of $\mathscr B$ to
depend on time, and do not impose any restriction on the period $T$ nor on the
averaged velocity, $\bar{\bfxi}$, of $\mathscr B$. If $\bfxi\not\equiv\0$ we
assume that $\bfxi$ and $\bfomega$ are both parallel to a constant direction,
while no further assumption is needed if $\bfxi\equiv\0$. We also furnish the
spatial asymptotic behavior of the velocity field, $\bfu$, associated to such
solutions. In particular, if $\mathscr B$ has a net motion characterized by
$\bar{\bfxi}\neq\0$, we then show that, at large distances from $\mathscr B$,
$\bfu$ manifests a wake-like behavior in the direction $-\bar{\bfxi}$, entirely
similar to that of the velocity field of the steady-state flow occurring when
$\mathscr B$ moves with velocity $\bar{\bfxi}$.
