  We propose a definition of tropical linear series that isolates some of the
essential combinatorial properties of tropicalizations of
not-necessarily-complete linear series on algebraic curves. The definition
combines the Baker-Norine notion of rank with the notion of tropical
independence and has the property that the restriction of a tropical linear
series of rank r to a connected subgraph is a tropical linear series of rank r.
We show that tropical linear series of rank 1 are finitely generated as
tropical modules and state a number of open problems related to algebraic,
combinatorial, and topological properties of higher rank tropical linear series
