  The most fundamental notion in frame theory is the frame expansion of a
vector. Although it is well known that these expansions are unconditionally
convergent series, no characterizations of the unconditional constant were
known. This has made it impossible to get accurate quantitative estimates for
problems which require using subsequences of a frame. We will prove some new
results in frame theory by showing that the unconditional constants of the
frame expansion of a vector in a Hilbert space are bounded by
$\sqrt{\frac{B}{A}}$, where $A,B$ are the frame bounds of the frame. Tight
frames thus have unconditional constant one, which we then generalize by
showing that Bessel sequences have frame expansions with unconditional constant
one if and only if the sequence is an orthogonal sum of tight frames. We give
further results concerning frame expansions, in which we examine when
$\sqrt{\frac BA}$ is actually attained or not. We end by discussing the
connections of this work to {\it frame multipliers}. These results hold in both
real and complex Hilbert spaces.
