I’m sorry, this is a long one. Unfortunately, the topic is complicated and I try to make it as understandable as possible. I’ll omit most of the technical details since they are hard to explain and don’t add a lot.
Executive Summary (“tl;dr”)
Epigenetic modifications do not constitute, nor enable, Lamarckian inheritance. Papers and reviews that claim this use a convoluted and ultimately useless definition of Lamarckism and inheritance.
However, a paper from December 2011 does actually demonstrate a very specific case where Lamarckian inheritace does occur (without epigenetic involvement). It bears repeating that this does not in the least violate Darwinism. In fact, the results, though surprising, are entirely in line and predicted by Darwinism – the mechanism actually relies on Darwinism. Contrary to popular belief, Darwinism and Lamarckism are not actually in opposition.
The problem, as so often, is the unclear definition of terms: what, exactly, is Lamarckian inheritance? Ask five scientists today and you will get six answers. And Lamarck himself would have disagreed with all of them. This is simply due to the fact that Lamarck wasn’t aware of a lot of mechanistic concepts that were developed long after his death – such as the distinction between somatic cells and germline.
This distinction leads to the concept of the Weismann barrier and it’s of eminent importance for our modern understanding of inheritance (and hence Darwinism): simply put, the Weismann barrier postulates that in order for traits to be inheritable, they need to be carried in the germline. As a consequence, modifications in the soma (non-germline cells) are not inherited.
Most scientists today see this as a direct opposition to Lamarckism.
But there are of course plenty of organisms (most, in fact) that do not have a distinction between soma and germline: notably, all unicellular life. For those life-forms, all genetically acquired traits (be it through mutation or through the integration of plasmids) are heritable. This is trivially true.
Once again, there are different definitions, but the generally agreed-on definition of epigenetics today runs something like this:
Epigenetics is the study of heritable changes in cell behaviour that are not due to changes in the genetic sequence.
Notice that here, “heritable” does not refer to heritability of organisms, just of cells: the daughter cells of a dividing mother cell inherit its epigenetic traits (and once again, this isn’t Lamarckism, it’s not even Darwinism, it’s on a much lower level).
There are several known mechanism of epigenetics, notable among them methylation and histone modification. How exactly they work isn’t terribly relevant; what’s important is that they modify the accessibility of genes on the genome: they control whether a cell can actually read a section of the genome and use it.
Cells use this to determine their specificity: how do muscle cells and brain cells know which job to do? Epigenetic modifications are (part of) the answer.
How Heritable Are Epigenetics?
Epigenetics are not generally heritable (due to the Weismann barrier). There have been several papers that, using similar lines of reasoning, have tried to argue that epigenetics can lead to the inheritance of acquired traits.
I will showcase the argument using one well-known paper, Epigenetic programming by maternal behavior .
The paper examined two different groups of mice which were raised by two phenotypically different mothers. Group 1 had been licked and groomed by their mother in the first six days after their birth. Group 2 hadn’t.
Female group 1 mice predominantly developed into mothers which would themselves lick and groom their offspring (“high LG/ABN”). Group 2 females developed into mothers which neglected their offspring (“low LG/ABN”), as shown in the following schematic:
It was shown that the mechanism behind this is epigenetic: pups of a high LG/ABN mother had increased serotonin levels which in turn led to decreased methylation of some genetic regions which modified their behaviour. And since the behaviour is self-perpetuating, the authors argued that it’s a form of inheritance (“soft inheritance”).
However, this is in fact not entirely true: the “inheritance chain” could be broken simply by raising the pups away from their mother for the first six days after birth. “True” inheritance would’t be broken by that. In fact, the whole argument is similar to saying that nationality is passed on by Lamarckian inheritance because children raised by, say, French parents are themselves predominantly French, and raise French kids.
Other papers (such as ) fall prey to similar flaws. Jerry Coyne has aptly summarised the whole field as follows:
In nearly all of these examples, the changes disappear after one or two generations, so they couldn’t effect permanent evolutionary change. […] I am not aware of a single case in which an adaptive change in an organism – or any change that has been fixed in a species – rests on inheritance that is not based on changes in the DNA.
This is pretty damning: none of these papers demonstrate what could reasonably be termed Lamarckism.
In December 2011, there has finally been a paper which convincingly demonstrated a real case of Lamarckian inheritance: Transgenerational Inheritance of an Acquired Small RNA-Based Antiviral Response in C. elegans .
The paper examines C. elegans (a nematode or roundworm which serves as a model organism in biology). Normally, those nematodes don’t get infected by viruses because they have developed a highly sophisticated defence mechanism called RNA interference (RNAi). This mechanism relies on so-called small-interfering RNAs (siRNAs) produced by the cell which recognise and pair up with viral RNA that has entered the cell. Those complexes are then in turn destroyed by the cell.
The paper looked at nematodes where the RNAi mechanism had been knocked out (removed): the cells could no longer generate siRNAs, and hence had no defence against the virus invaders. Now, if you artificially put siRNAs into the cells, those cells can once again defend against a specific virus. The authors could show that under very specific circumstances, putting siRNAs into the cells was enough to trigger an inheritable virus resistance, even though the siRNA-generating mechanism was still knocked out.
The nematodes had inherited an acquired, non-genetic trait (virus resistance), and this inheritance was sustained over many generations.
Since I’ve explicitly mentioned the Weismann barrier before, I should note that this experiment does not invalidate the Weismann barrier: this heritability only occurred in very specific circumstances, namely, when all of the cells in the nematode were provided with siRNAs. This notably includes the germline cells.
Despite frequent claims to the contrary, no connection between epigenetics and Lamarckism has ever been demonstrated. Inheritance of acquired traits from epigenetic modifications is transient at best, and it’s contested whether it constitutes inheritance at all.
On the other hand, a clear example of Lamarckism has now been observed, albeit under very constrained, artificial circumstances. And while this is a very interesting result, it is not surprising: any competent biologist would have predicted the paper’s result, given the paper’s premises, and ignoring potential ways in which the experiment could go wrong, because its outcome is entirely predicted by modern biology.
A Final Remark
There are a lot of misconceptions flying around. Articles with titles like Why everything you've been told about evolution is wrong abound. There is no nice way to put it – these pronouncements are untrue. Nothing found in the papers cited here, nor in similar papers, either directly or implicitly contradicts “conventional evolution” – Darwinism. In fact, all of these studies heavily rely on predictions made by modern evolutionary theory. It’s just blatant misreporting by the media, due to two unfortunate facts: most science journalists don’t understand the science, even rudimentarily; and most scientists can’t even explain their research clearly to their colleagues, let alone lay people.
 Ian C. G. Weaver, Nadia Cervoni & al., Epigenetic programming by maternal behavior, Nature Neuroscience 7, 847–854 (2004)
 Sheau-Fang Ng, Ruby C. Y. Lin & al., Chronic high-fat diet in fathers programs β-cell dysfunction in female rat offspring, Nature 467, 963–966 (2010)
 Oded Rechavi, Gregory Minevich & Oliver Hober, Transgenerational Inheritance of an Acquired Small RNA-Based Antiviral Response in C. elegans, Cell 147 (6), 1248–1256 (2011)