  We systematically analyze the operator content of unitary superconformal
multiplets in $d > 3$ spacetime dimensions. We present a simple, general, and
efficient algorithm that generates all of these multiplets by correctly
eliminating possible null states. The algorithm is conjectural, but passes a
vast web of consistency checks. We apply it to tabulate a large variety of
superconformal multiplets. In particular, we classify and construct all
multiplets that contain conserved currents or free fields, which play an
important role in superconformal field theories (SCFTs). Some currents that are
allowed in conformal field theories cannot be embedded in superconformal
multiplets, and hence they are absent in SCFTs. We use the structure of
superconformal stress tensor multiplets to show that SCFTs with more than 16
Poincar\'e supercharges cannot arise in $d \geq 4$, even when the corresponding
superconformal algebras exist. We also show that such theories do arise in $d =
3$, but are necessarily free.
