Heredity, or the passing of traits from parents to offspring, was long thought to occur exclusively through the transmitting of DNA from one generation to the next. Recently, however, there’s been somewhat of a change as the possibility of non-genetic inheritance (that is, through things other than DNA) has been shown. A recent article discusses how the concept of heredity has evolved, and how non-genetic mechanisms of inheritance might affect it.
Beginning in the 18th century, the scientific investigation of heredity led to the emergence of two schools of thought around the late 19th century: the soft and the hard one. Soft inheritance means that traits the parents acquired during their lifetime could be transmitted to their offspring. A well-known proponent, often mentioned in this context, was Jean-Baptiste de Lamarck, a French naturalist. A famous example of what is sometimes referred to as Lamarckism, is the long neck of giraffes. According to soft inheritance, giraffes stretch out their necks in order to reach food, thereby slightly lengthening their neck (thus, an acquired trait). Their offspring will inherit this and have longer necks.

The evolution of the giraffe’s neck explained by Lamarck (top) and Darwin/Wallace (bottom).
(Source: http://www.biologycorner.com/APbiology/evolution/ch17.html)
But, since the Modern Synthesis in evolutionary biology in the first part of the 20th century, this idea was refuted, and hard inheritance was proposed as sole mechanism of heredity. This means that, at the moment of conception, the parents transmit the genetic blueprint to their offspring. So, the use or disuse of a trait by the parents does not affect this trait in the offspring.
Recently, however, non-genetic mechanisms of inheritance have been and are being uncovered, ranging from epigenetic inheritance to its cultural cousin. This suggests that there is more to it than just DNA after all. However, this does not necessarily engender a return to the old notion of soft inheritance, but rather implies the development of a pluralistic model of heredity. This is shown in the following figure, where (a) represents the hard stance where all that matters is DNA, (b) the genetic encoding model of soft inheritance, wherein the environment affects the body (soma), which in turn affects the DNA, and (c) the pluralistic model where both DNA and non-genetic factor (which can be influenced by the environment) play their part.
The author concludes:
… recent challengers of the established view of heredity propose a pluralistic model, whereby heredity reflects the transmission of gene alleles (genetic inheritance) alongside a variety of other factors that influence offspring phenotype (nongenetic inheritance). The pluralistic model of heredity points to the existence of variation in the nature of inheritance among taxa and traits. Although the existence of nongenetic inheritance is not in doubt, some putative mechanisms of nongenetic inheritance remain poorly understood, and controversy persists over the role and importance of nongenetic inheritance in shaping phenotypic variation and influencing the dynamics and course of evolution.
So, the debate is not about whether or not non-genetic inheritance happens, but rather about how important it is (or can be) in the process of evolution.
(I’ve added two additional references below (Bonduriansky et al., 2011 & Danchin et al., 2011) of reviews concerning non-genetic mechanisms of inheritance, discussing examples, and potential future avenues of research. You should be able to find pdf’s of these online.)
Also, Cell Press has a discussion page devoted to the article, where several heavyweights in the field have already expressed their opinion.
(!) P.S.:
Non-genetic inheritance (and certainly epigenetics) is sometimes, sadly, used against Darwinism. This is simply wrong. Darwin, in his day, was unaware of the currently known mechanisms of inheritance (DNA and the recent non-genetic complements). The suggestion that there might be more than one such mechanism does in no way disprove the Darwinian idea of evolution.
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Bonduriansky, R. (2012). Rethinking heredity, again Trends in Ecology & Evolution, 27 (6), 330-336 DOI: 10.1016/j.tree.2012.02.003
Additional references:
Bonduriansky, R., Crean, A., & Day, T. (2012). The implications of nongenetic inheritance for evolution in changing environments Evolutionary Applications, 5 (2), 192-201 DOI: 10.1111/j.1752-4571.2011.00213.x
Danchin É, Charmantier A, Champagne FA, Mesoudi A, Pujol B, & Blanchet S (2011). Beyond DNA: integrating inclusive inheritance into an extended theory of evolution. Nature reviews. Genetics, 12 (7), 475-86 PMID: 21681209
