How to account for publication bias? I am doing a literature review in the medical field, which is as many other fields heavily impacted by publication bias. Is there any standard way to account for it? 

When am I e.g. allowed to say that treatment X successfully treat condition Y, despite the fact that a certain amount of studies may have been left unpublished as they did not show statistically significant "positive" results, i.e. that those unpublished studies did not successfully show that treatment X does treat condition Y?

Or shall I just ignore it?
First, you should note the difference between checking for evidence of publication bias and correcting for it. To know that there are patterns in the data that indicates publication bias means that you will be able to temper the discussion of your results, and maybe suggest potential weaknessess in the study, even if it is not possible to correct for biases.

There exists a number of methods that can be used to identify publication bias. A simple graphical funnel plot of effect size vs. sample size is maybe the easiest to apply. The idea here is that the scatter around the true effect size should be symmetric and with expanding variance at smaller sample sizes ("trumpet-shaped"). Publication bias can lead to underreporting of non-significant studies with smaller sample sizes, which would result in a skewed pattern in the funnel plot. Another method to test for publication bias is a regression between standardized effects and precision ('Egger's method') or rank-correlation methods. You can get an overview of these methods and others in e.g. Macaskill et al. (2001) and Thornton &amp; Lee (2000).

Accounting/correcting for publication bias is trickier, and I have no personal experiance in using these methods. However, I know that one method that is used in medical research is to only include pre-registred studies in the meta-analysis. If this is possible depends on your topic though. You can also use simulations and parametric methods to basically try to recreate the complete (unknown) dataset, and in this way account for the publication bias (one example of this is Givens et al. 1997)