A central task in evolutionary biology is the construction of phylogenetic trees to represent the ancestral history of a collection of present-day taxa. The data typically used for this task are characters, which describe the attributes of the taxa under consideration. To illustrate, a particular position of aligned DNA sequences is an example of a character. The attributes (states) of this character are the subsets of taxa taking a certain nucleotide at this position. In the context of this task, many questions naturally arise. How do we decide if a collection of characters is compatible (realisable by a single tree)? Does compatibility depend on considering all of the characters in the collection simultaneously or can we do better than that? How many characters does it take to uniquely determine a phylogenetic tree? Does the answer depend on the topology of the tree? If so, how? If the characters have a bounded number of states, how does the number of characters it takes grow with respect to this bound and the number of taxa? In this talk, we investigate these questions, and discuss some old and not so old results.