How to define stationary sets in general If k is a regular, uncountable cardinal, then we call S $\subseteq$ k stationary if, and only if, S meets every closed unbounded subset of k.  Can we use the same definition, word for word, to define stationary sets even if k is an arbitrary limit ordinal?
 A: Sure, but this is only meaningful if $k$ has uncountable cofinality. 
Otherwise, if $k$ has countable cofinality, we do not have that the intersection of club sets is club (any cofinal $\omega$-sequence is club), and the stationary sets are just the complements of bounded sets. 
But for ordinals of uncountable cofinality, the notion of stationary set makes sense and is useful. Of course, one needs to be careful when generalizing the standard results to this setting. For example, Fodor's lemma needs a small adjustment. 
By the way, this is just one of the ways we can generalize the notion of stationarity. One generalization considers not ordinals but sets of the form $\mathcal P_\kappa(\lambda)=\{X\subset\lambda\colon |X|<\kappa\}$, and yet another considers arbitrary sets. Both have applications in different settings. Jech's paper in the Handbook discusses some of the properties of these generalized versions.
A: As Andres said, your proposed definition is fine if $k$ has uncountable cofinality, so the only question is what, if anything, to do with $k$ of countable cofinality.  My preferred formulation is that a set is stationary if it meets every set in the filter generated by the closed unbounded (= club) sets.  When $k$ has uncountable cofinality, this agrees with what you wrote, because the intersection of any two clubs is a club.  When $k$ has countable cofinality, then there are two disjoint clubs, so the filter generated by clubs is the improper filter --- it contains every subset of $k$, even the empty set --- and therefore no set, not even all of $k$, is stationary.  In the (admittedly few) situations where I found it convenient to have a general definition of stationarity for all limit ordinals, this convention, that no set is stationary in an ordinal of countable cofinality, worked smoothly. 
