Pseudogenes are genes present in an organism's genome that have lost the ability to code for proteins due to mutation.  They were first identified and dubbed in the late 1970s when researchers began finding non-coding regions in some organisms that were similar to actual coding genes in other organisms.  So far an estimated 19,000 pseudogenes have been identified in the human genome, this is almost equal to the total number of coding genes (21,000).  Pseudogenes have been identified in a wide range of organisms from bacteria to mice to humans, the total number of pseudogenes in a given genome is not predictable but specific pseudogenes are often compared across species to elucidate complex evolutionary relationships  Humans have many pseudogenes including L-gulonolactone oxidase which is used to synthesize vitamin c. This gene was inactivated in the common ancestor of all simians.
Pseudogenes are often difficult to parse from the large amount of non-coding base pairs in the genome. Convention requires two elements to be present to label a sequence a pseudogene. The first is homology which is the requirement that a sequence be demonstrated to descend from a functional copy of the gene and the second is non-functionality which is the requirement that the gene not code for a protein in the organism in question. 
Since all pseudogenes are descended from a functioning gene the first step is to find the parent gene that it descended from. This is done by using computer programs to compare sequences of DNA across species.  This is a large computational problem but by keeping in mind the phylogenetic relationships between species the search time can be decreased by looking at species that share a more recent common ancestor. Once a functioning copy of a gene is detected its sequence is compared to the pseudogene. A high correlation in base pairs is used to assign homology. Non-functionality can be demonstrated by attempting to transcribe the sequence in-vitro. 
Pseudogenes and neutral selection theory
Because pseudogenes do not code for a function many scientist have hypothesized that the accumulation of mutations would not be constrained by selection pressures. This is known as neutral selection, and pseudogenes have been studied extensively to test various theories of neutral selection.  It has been determined that mutations fixate in pseudogene regions at about 30 percent higher than in coding regions of DNA.  Some theorist have argued that there maybe some selection pressure on pseudogenes (such as on genome size in general) so conclusions should be tempered. Others have determined that base pair mutations are not completely random, favoring accumulation of guanin and cytosine.  Despite these findings research on pseudogenes still continues to be a productive avenue for exploring mutation and selection.
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