There is no irrefutable proof that smoking causes cancr, a “fact” which the tobacco companies are fond of reiterating.
There is, however, a strong correlation between smoking and lung cancer. Repeated studies have shown that smokers are about ten times more likely to get lung cancer than non-smokers. The first major study to prove the link was published in Britain in 1950 by Doll and Hill (Cigarette smoking and lung cancer (Doll and Hill British Medical Journal 1950 ii 739-748). I can’t find a copy of the paper online, but given the cite, somebody else might be able to.
Lung cancer is very rare in the non-smkoing population, so even if you multiply the risk by ten, smokers are still relatively unlikely to die of lung cancer. They are much more likely to die of coronary artery disease: although the relative risk of coronary artery disease associated with smoking is only about 1.7 (i.e. a smoker is 1.7 times more likely to get it than a non-smoker), it is much more common in the general population. It’s a small but significant increase to a big risk, rather than a big increase to a small risk.
Broadly speaking, if you take a large group of smokers of various ages and compare them to an age-matched group of non-smokers (i.e. you have to have the same number of 30 year-olds, 76 year-olds, etc. in each group), the relative death rate of the smokers is about 1.6. Put crudely, a smoker is 50% more likely to die at any given age than a non-smoker (though obviously they’ll both die eventually).
Now, it could be that the gene which predisposes some people to smoke also predisposes them to get cancer and heart disease, though there’s no evidence for this. It could be that “smoking-related” diseases are in fact related to poverty (poor housing, diet, etc.) and that a greater proportion of poor people than rich people smoke. There is some evidence for that, but the risks associated with smoking were first established in the 1950s when smoking was much less a function of social class. It could be that the risks associated with smoking are not to do with smoking per se, but come from the chemicals in matches or lighter fuel (no evidence for this either).
It could even be, as others have suggested, blind chance. All those studies into smoking and lung cancer, involving hundreds of thousands of people (if not millions) could still have inadvertantly selected groups of smokers who were more prone to lung cancer anyway. Likewise, it could be pure coincidence that, up till now, things have always moved downwards, rather than upwards, when dropped. Or that the Sun has always risen in the morning and set at night.
In other words, it depends what you mean by “proof”. On the skeptical/Popperian view of science, nothing is ever proven. For classical skeptics (e.g. David Hume), there is simply no proof there to be had; what scientists term “causation” is merely “constant conjunction”. For Popper, there is only failure to disprove, though it’s difficult to see how you could disprove the hypothesis that smoking leads to an incerased risk of lung cancer: Popper’s model works quite well with yes/no type questions and less well with relative risk. (Popper’s model also, incidentally, comes so far short of describing how scientists actually work as to be laughable.)
If by “proof” you mean the generally-accepted standards of medical proof: that is, the kind of steps you would have to go through to prove that a drug was safe and effective before it could be licenced, then the proof is pretty irrefutable: if you compare large numbers of smokers with large numbers of non-smokers, then the smokers are ten times more likely to get lung cancer. It’s as simple as that.
