Is String Theory proven to be finite?

A commented list of literature with claimed results on (super-)string perturbative finiteness is here:

Notice the technical caveats in remarks 1 and 2 at the beginning of this entry.

In summary the statement is: there are plenty of arguments that the (super-)string is UV-finite at each order and this argument is regarded as robust. There are much more recently only computations of the actual integrals over (super-)moduli space which also come out finite (hence IR finite) but which have been done in detail only at low loop order (since this is technically much more demanding). Arguments by Berkovits that the pure spinor formulation helps here seem to have not been further followed up much (?).

One issue apparent from the list of literature is that theoretical physics is suffering here a bit from its lack of mathematical certainty: it is not always clear whether a claimed result has really been established, or just made very plausible, and what exactly has been claimed. For instance often one sees people point to Madelstam's article (listed at the above link) as a proof of finiteness, while Mandelstam himself, according to his Wikipedia article, says he only showed the absence of one of several possible divergences.

It is perhaps a good idea to answer this question (6 years later) by pointing new exciting developments about how precisely string theories avoid perturbative inconsistencies.

The key property of the perturbative string finiteness is the UV/IR connection. I strongly recommend Ultraviolet and Infrared Divergences in Superstring Theory to gain an intuition of this connection. After the identification of UV divergences as IR effects, soft theorems are needed to demostrate that the IR divergences can be cured Of course the latter is subtle in perturbative string theory (where adjectives like "soft" and "off-shell" are a little bit mysterious). It is convenient to highlight the outstanding String Field Theory as World-sheet UV Regulator. I am not aware of any other beautiful application of string field theory to ordinary perturbative string vauca of this type. A truly lovely paper that rigorously exhibit the perturbative healthy of string theory.

I'm also amazed that nobody has mentioned section 9.5 of Polchinski string theory (Vol. 1) textbook. Where higher-genus amplitudes and degenerate worldsheet contributions are analyzed in detail.