One common, but sometimes difficult, question is how best to decide which species an organism belongs to, because reproductively isolated groups may not be readily recognizable, and cryptic species may be present. There is a continuum from reproductive isolation with no interbreeding, to panmixis, unlimited interbreeding. Populations can move forward or backwards along this continuum, at any point meeting the criteria for one or another species concept, and failing others.
The current meaning of the phrase "species problem" is quite different from what Charles Darwin and others meant by it during the 19th and early 20th centuries. For Darwin, the species problem was the question of how new species arose. Darwin was however one of the first people to question how well-defined species are, given that they constantly change.
... I was much struck how entirely vague and arbitrary is the distinction between species and varieties