In a recent talk at the UCLA Logic Colloquium, Martin Davis reported that Kurt Godel at first suspected that V=L might be a reasonable axiom, but later came to believe that the continuum hypothesis was false. Many mathematicians since have suggested this also, including Cohen, Shelah, and Woodin. The question continues to occupy a central place in modern set theory.
In the paper Remarks on Levy's Ref;ectiom Axiom (PDF) in the 1993 Mathematical Logic Quarterly, I argued that CH was true, a position I came to in 1985. A recent letter in the AMS Notices suggested that there might not be a simple argument deciding CH; my article was not mentioned. I sent a letter to the AMS Notices pointing it out; they declined to publish the letter on the grounds of "self promotion".
As a public service a simple argument are given here. The most succinct statement is, that the independence of the existence of an injection of Aleph_2 into P(omega) is strong evidence that no such embedding exists. Being a computer scientist, I arrived at this observation by a constructive method. The embedding of Aleph_1 in P(omega) was discovered by Cantor. Attempts to extend this to Aleph_2 fail; and indeed by Godel's consistency theorem are doomed to.
I have since run across this argument on another web page, but it does not seem to be particularly well-known.
A variety of inter-related arguments can be given that CH is true and V=L. This might be called the "minimalist" view, in contrast to the currently more widely held "maximalist" view that quite large cardinals exist, and regularity holds for a large collection of sets of reals. One discussion can be found in the above cited article. A more recent discussion can be found in "Iterating Mahlos operation", IJPAM 9 (2003). This is ongoing research, and a new manuscript is in preparation.