  Ryser's conjecture says that for every $r$-partite hypergraph $H$ with
matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$.
This far reaching generalization of K\"onig's theorem is only known to be true
for $r\leq 3$, or $\nu(G)=1$ and $r\leq 5$. An equivalent formulation of
Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with
independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$
monochromatic connected subgraphs which cover the vertex set of $G$.
  We make the case that this latter formulation of Ryser's conjecture naturally
leads to a variety of stronger conjectures and generalizations to hypergraphs
and multipartite graphs. Regarding these generalizations and strengthenings, we
survey the known results, improving upon some, and we introduce a collection of
new problems and results.
