Figure 1. A comparison of the number of times each term ("degeneracy", "redundancy", and "robustness") appears in the scientific literature.
Evidence for degeneracy can be found in the existence of multiple routes to a specific physiological function. According to , members of the adhesins gene family in yeast can interchangably perform functional roles when expression is elevated. This is in contrast to lock-and-key systems (e.g. receptor-ligand binding), which interact with a high degree of specificity . Another common signature of degeneracy is cross-talk, which is common in gene regulatory and neural pathways. The presence of degenerate pathways (or at least the name for them) suggests a "degeneration" from some previous state. This is at least partially correct: it appears that degenerate relationships involve a generalization of function over evolutionary time. However, this need not result from a mechanism that was originally functionally specialized.
To understand this in context, let us return to the "lock-and-key" model. Lock-and-key systems are highly specialized with regard to function. In fact, if one were to consider only the end product, we might be tempted to conclude a purposeful design. However, if we consider that both the lock AND key have shared evolutionary histories, it becomes more possible that this arrangement is not only the product of mutation-selection dynamics but historical contingency as well . Lock-and-key phenomena result from selection for extreme specialization . While this might have a fitness advantage in some contexts, in highly-veriable environments it is not particularly advantageous. Thus, the historical lock-in  that inadvertently results from selection can results in evolutionary dead-ends. What degeneracy provides, then, is a means to either rescue a phenotype from or circumvent entirely such instances.
There are also potential evolutionary tradeoffs between network functionality and function of the individual components of this network. One example of this is the role of an individual gene in a genetic regulatory network. Whether degeneracy result from a true evolutionary tradeoff or as a signature of a complex system's emergent properties  is not clear. However, Whitacre and Bender  modeled biological networks as a complex adaptive system (CAS). Using this approach, robustness was found to result from both diversity and degeneracy. In this case, invariance to perturbation (e.g. robustness) results from many possible ways to achieve a common function. Diversity allows for the biological system to move away from the extreme specificity required of the "lock-and-key" model , while a degenerate architecture incorporates this diversity into a functional mechanism.
Figure 2. A schematic showing the relationship between degeneracy and the related concepts of complexity, robustness, and evolvability. Adapted from Figure 1 in .
Tononi, Sporns, and Edelman  have proposed ways to quantify degeneracy in biological networks. The quantification is based on the notion that redundancy and degeneracy stand in contrast to all outputs of a system (e.g. gene network) being statistically independent of one another. The concept of mutual information  is used to quantify the degree of a shared functional role between output. If two or more output share information, the related functions are said to exhibit degeneracy. Another quantitative approach is to think of degenerate biological systems as degenerate sets of overlapping functions [1, 13]. Given its mathematical similarity to phylogenetic theory, such an approach might reveal new insights into convergent evolution.
What can degeneracy teach us about complex biological systems? One lesson is that in some cases there may be a fitness benefit for maintaining a parallel architecture. While not clearly beneficial in every context, parallelism can perform critical functions such as buffer against mutations or act as a noise filter . A second lesson is that degeneracy is not always degenerate: far from being a failure of optimization, degeneracy provides a means to incorporate the stuff of evolutionary time (mutation) into a system that does not become reliant on any single pathway (specificity). In this way, degenerate biological systems are often the most adaptable, which means that some outcomes of the evolutionary process can truly be described as "survival of the most degenerate".
 Edelman, G.M. and Gally, J.A. Degeneracy and complexity in biological systems. PNAS, 98(24) 13763–13768 (2001).
 Whitacre, J. and Bender, A. Degeneracy: A design principle for achieving robustness and evolvability. Journal of Theoretical Biology 263 (2010) 143–153.
 Data for graph courtesy of PubMed, search data August 27, 2012.
 For concept of historical contingency, please see: Swartz, B.A. On the “Duel” Nature of History: Revisiting Contingency versus Determinism. PLoS Biology, 7(12), e1000259 (2009). AND Fontana, W. and Schuster, P. Continuity in Evolution: On the Nature of Transitions. Science, 280(5368), 1451-1455 (1998).
 References to historical lock-in can be found in: Nelson, R. and Winter, S. An evolutionary theory of economic change, Harvard University Press, Cambridge, MA (1982). This term is often used in the economics and business literature, but also has relevance to the biological world.
 Whitacre, J.M. Degeneracy: a link between evolvability, robustness and complexity in biological systems. Theoretical Biology and Medical Modeling, 7(6), 6 (2010).
 A system with emergent properties produces an output which is greater than the sum of its parts. Weak emergence is a case where the collective effects can be reduced to its individual components, while strong emergence results in collective effects that are irreducible. In many cases, biological evolution can be thought of as resulting from strong emergence.
For information specific to evolution, please see: Blitz, D. Emergent Evolution: Qualitative Novelty and the Levels of Reality. Kluwer Academic, Dordrecht (1992) AND Bedau, M.A. Downward causation and autonomy in weak emergence. Principia, 6, 5-50 (2003).
 Whitacre, J.M. and Bender, A. Networked buffering: a basic mechanism for distributed robustness in complex adaptive systems. Theoretical Biology and Medical Modeling, 7, 20 (2010).
* the complex adaptive systems (CAS) approach is a way to model systems of high complexity using a series of interacting agents that originated out of the Santa Fe Institute.
For a general overview, please see: Holland, J.H. Studying Complex Adaptive Systems. Journal of Systems Science and Complexity, 19, 1–8 (2006) AND Miller, J.H. and Page, S.E. Complex Adaptive Systems: An Introduction to Computational Models of Social Life. Princeton University Press, Princeton, NJ.
 Gomez-Gardenes, J., Moreno, Y., and Floria, L.M. On the robustness of complex heterogeneous gene expression networks. Biophysical Chemistry, 115, 225-228 (2005).
 Tononi, G., Sporns, O., and Edelman, G. Measures of degeneracy and redundancy in biological networks. PNAS, 96, 3257-3262 (1999). This work was developed for studying brain networks, but theoretically can be applied to a wide range of biological systems, including genetic regulatory networks.
 For a general introduction, please see: Latham, P.E. and Roudi, Y. Mutual information. Scholarpedia, 4(1), 1658 (2009). Link.
 I could find no references to this outside of the original paper (Edelman and Gally). I imagine it combines the mathematical concept of degeneracy (when objects change their set membership over time) and conventional set theory.
 For mutational and phenotypic buffering, please see: Braendle, C. and Felix, M.A. Plasticity and Errors of a Robust Developmental System in Different Environments. Developmental Cell, 15(5), 714-724 (2008) AND Rutherford, S.L. and Lindquist, S. Hsp90 as a capacitor for morphological evolution. Nature, 336-342 (1998).
For noise filtering in a genetic regulatory network, please see: Orrell, D. and Bolouri, H. Control of internal and external noise in genetic regulatory networks. Journal of Theoretical Biology, 230(3), 301-312 (2004) AND Lestas, I., Paulsson, J., Ross, N.E., and Vinnicombe, G. Noise in gene regulatory networks. IEEE Transactions in Automation and Control, 53, 189-200 (2008). .
** For the latest information on degeneracy research, please see the work of James Whitacre, admin of the Degeneracy and Selection online community. Also, the Wikipedia page for Degeneracy (biology) features a comprehensive bibliography featuring papers from many areas of biology.