The primary suspect was a man named Arthur Leigh Allen, based largely on circumstantial evidence (he owned the same kind of typewriter used to type the famous letters, had a Zodiac brand wristwatch-significant not just for the name, but also because the brand logo is the same crosshairs design the killer used repeatedly at crime scenes and in letters-, lived very close to the first victim, had a criminal history, matched the physical description height and weight wise given by survivors; most damning of all, he had a conversation with a friend during the course of a weekend fishing trip in which he outlined a fantasy that mirrors the Zodiac crimes, this only really being damning assuming you believe the friend) and advocated by Robert Greysmith (writer who has published frequently about the Zodiac Killer). ALA is the suspect advanced/implied in the film "Zodiac", which came out a few years back. No conclusive evidence was ever found that linked him to the killings, and he died in the 1990's (1992). DNA testing done in the early 2000's demonstrated that ALAs DNA was not on the envelopes or letters sent to the San Francisco Chronicle by the killer, however there is also no way of knowing if the killer's DNA was ever actually on the letters (he could have worn gloves, dozens of people had handled the letters prior to testing, etc.). So this DNA test did not exclude him as the killer, it just excluded him as someone who left DNA on the letters in some capacity. As far as I know he is still considered a viable suspect, but the evidence remains inconclusive. Another thing to consider about ALA is that handwriting experts poured over writings confirmed to be his and found absolutely no stylistic similarities to the Zodiac letters... so it's up in the air.

Besides ALA there have been a number of other suspects suggested with varying levels of plausibility, but all have been advanced based on circumstantial evidence or second-hand testimony (i.e. "my dad's brother's friend was the Zodiac"). Other well known killers have even been suggested, including The Unabomber (now discounted). Two recent developments include a San Francisco lawyer coming forward to claim that a merchant seaman came into his office one night, confessed to the killings, and then vanished again (a scenario regarded as plausible by police and authorities on the case); additionally, police are looking for a man present in a 1960's photograph with one of the known victims who matches the description of the suspect, but as far as I know he has yet to be ID'd/found. The case remains open in multiple jurisdictions in California, and the FBI maintains an active investigation.

Source: Robert Greysmith's book "Zodiac", Wikipedia (http://en.wikipedia.org/wiki/Zodiac_Killer#Suspects), amateur interest/reading on the subject over the years. I'd really like to hear the perspective of someone in law enforcement about the possibilities of solving this case, etc.

Edit: There is a ton of information here (http://www.zodiackiller.com/AllenFile.html) about ALA and the various things linking him to the killings, though it is still all circumstantial... but some of it is pretty damning. After reading RG's book I was pretty convinced there was at least a strong case against him.