One contrasts two natural measures of the complexity of a 3-manifold, namely the rank of the fundamental group and the Heegaard genus. The Heegaard genus is an upper bound for the rank and F. Waldhausen in 1967 was apparently the first to ask in print if the rank and Heegaard genus could actually differ. Waldhausen asked the question for closed manifolds, but there has been considerable interest in the question for bounded manifolds, where Heegaard genus is usually replaced with the dual notion of tunnel number. Some of the questions early on in R. Kirby's famous 1976 list turn out to be just special cases of this question. A particularly sharp form of Waldhausen's question is the "Meridional generator conjecture" which in essense asserts that "meridional generators" exactly correspond to "bridges". Recent progress on this and on other forms of Waldhausen's question will be discussed herein.