What's in a name?
Need a definitive 'take' on which of Shakespeare's plays are romances, histories and just plain problems?
Well, join the club: we're pretty much all in the same boat when it comes to pinning down exactly what's what... and centuries of scholarship and evolving theories tend to complexify rather than secure the debate.
For me, The Royal Shakespeare Company is as good a place as any to start, and I use their list of comedies / tragedies / histories as the foundation for my own understanding, and you can access their page - which is packed with information about each individual play - here.
When speaking about Shakespeare's genres, these are the essential three to focus on. I suggest that you don't muddy the waters by delving into the sub-genres for pub quizzes / general knowledge purposes. Too many learners get confused with the idea of 'romances' or 'revenges' and forget to hone in on the three fundamental genres which best frame nascent understanding.
However, once you're fully secure in your understanding of which plays make up the comedy / tragedy / history canon you might be interested to look more deeply at the sub-genres that arise.
The 'problem' plays are generally accepted to be All's Well That Ends Well, Measure for Measure, and Troilus and Cressida: each one flipping between occasional broad comedy and dark psychological drama. F.S. Boas first coined the term 'problem play' in a thesis of 1896, and the term has come to describe either the content of the play, or a question over its classification.
Other critics include The Merchant of Venice, Timon of Athens, and The Winter's Tale within this 'problem' sub-genre, and it is not unusual for other plays to make an appearance depending on how their content is viewed through the prism of the time and social standpoint that they are being critiqued. That these 'problems' cross genres makes them doubly tricky to navigate, and a considerable amount of research is necessary before feeling confident in asserting that any individual play is indeed a 'problem'.
Another interesting sub-genre is the 'Roman' play, and these - by dint of their chronology - are somewhat easier to identify, although there are nevertheless intellectual arguments as to which plays qualify. Antony & Cleopatra, Coriolanus, Julius Caesar, and Titus Andronicus are generally accepted as the four Roman plays, given that they are set in Ancient Rome and all use Plutarch as their main source. However, some critics include Cymbeline in this sub-genre.
Late 'romances' were firstly categorised by Edward Dowden in 1875, and it is true that Pericles, Cymbeline, The Winter's Tale, The Tempest and The Two Noble Kinsmen have elements which lend themselves to this categorisation. These later plays tend to fall within the comedy genre, and almost all feature extravagant scenery and mystical representations which are thought to be have suited the audience expectations of the time. They may carry comic elements, but often focus on psychological drama and carry particular emotional resonances in their depiction of father / daughter relationships.
Pericles, Prince of Tyre and The Two Noble Kinsmen were not included in the First Folio collection of Shakespeare's plays in 1623. It is from the First Folio that we are given the concept of comedies, tragedies and histories. Personally, I don't count The Two Noble Kinsmen as part of the Shakespeare canon, although if someone were to present a piece from TTNK to me, I would absolutely accept it as a Shakespearean piece. I do however count Pericles... not that he ever seems to crop up all that much.
For me, it is essential that one roots personal understanding of Shakespeare's genres within the original comedy, tragedy, history framework: this then allows scope for personal determination - which must be suitably, critically and exhaustively justified - of sub-genres.