What are the main weaknesses of a Playfair cipher, if any? What are the main weaknesses of a Playfair cipher, if any?

I know that they depend on none of the letters missing, but that is an easy fix if a letter gets dropped. Besides that, are there any other problems with it?
It's a quite a weak cipher, being better than a simple substitution cipher by only using digraphs instead of monographs.

An interesting weakness is the fact that a digraph in the ciphertext (AB) and it's reverse (BA) will have corresponding plaintexts like UR and RU (and also ciphertext UR and RU will correspond to plaintext AB and BA, i.e. the substitution is self-inverse).
That can easily be exploited with the aid of frequency analysis, if the language of the plaintext is known.

Will chaining Playfair with a substitution cipher help?
Nope... The (monoalphabetic) substitution cipher will act (almost completely) transparent against the frequency attack. Eventually, the Playfair cipher is a digraph substitution cipher itself. (But it will make the resulting cipher not self-inverse anymore.)

Well, polyalphabetic ciphers are a whole different case of course. It's still a play-toy for today's computers, but it'd completely render the text unbreakable in that era. But then again, we needed machines like Enigma to properly and acceptably implement it. The German Enigma, if I'm not mistaken, implemented a polyalphabetic cipher.