  A connected covering is a design system in which the corresponding {\em block
graph} is connected. The minimum size of such coverings are called {\em
connected coverings numbers}. In this paper, we present various formulas and
bounds for several parameter settings for these numbers. We also investigate
results in connection with {\em Tur\'an systems}. Finally, a new general upper
bound, improving an earlier result, is given. The latter is used to improve
upper bounds on a question concerning oriented matroid due to Las Vergnas.
