  Weighted and controlled frames have been introduced recently to improve the
numerical efficiency of iterative algorithms for inverting the frame operator.
In this paper we develop systematically these notions, including their mutual
relationship. We will show that controlled frames are equivalent to standard
frames and so this concept gives a generalized way to check the frame
condition, while offering a numerical advantage in the sense of
preconditioning. Next, we investigate weighted frames, in particular their
relation to controlled frames. We consider the special case of semi-normalized
weights, where the concepts of weighted frames and standard frames are
interchangeable. We also make the connection with frame multipliers. Finally we
analyze weighted frames numerically. First we investigate three possibilities
for finding weights in order to tighten a given frame, i.e., decrease the frame
bound ratio. Then we examine Gabor frames and how well the canonical dual of a
weighted frame is approximated by the inversely weighted dual frame.
