In my opinion, it is hard  (or virtually impossible)  to define "species" with one definition, because it is not exact as math. There are several definitions fo "species" created by humans to establish a "picture" of organisms' groups that share characteristics or can reproduce and generate viable offspring at a given time. But this makes sense for us to study, to classify, taxonomically . But, in reality, this division does not exist, it  is a continuous phenomenom drived by evolution... It is like a movie, not a picture. Species are evolving and these artificial boundaries for establishing "species"  are being modified day by day. But it is a interest topic in Biology. I always like to discuss it. It is just one more opinion.

In my opinion, it is hard  (or virtually impossible)  to define "species" with one definition, because it is not exact as math. There are several definitions fo "species" created by humans to establish a "picture" of organisms' groups that share characteristics or can reproduce and generate viable offspring at a given time. But this makes sense for us to study, to classify, taxonomically . But, in reality, this division does not exist, it  is a continuous phenomenom drived by evolution... It is like a movie, not a picture. Species are evolving and these artificial boundaries for establishing "species"  are being modified day by day. But it is a interest topic in Biology. I always like to discuss it. It is just one more opinion.

Individuals exist, but species do not.

I advise reading up on "species concepts". There are over 13 I believe? The "biological species concept" refers to the definition of species being a group of individuals that can breed to produce viable offspring. However there are other species concepts that revolve more around molecular methods or other observations. It's a very interesting topic to read about! 

You'll find a lot of hybrids if you investigate plant genera.

Actually, even the definition of individual becomes fuzzy once you start looking at colonial and clonal species. Not to mention the idea of complex symbioses and microbiomes.

Asif,

Actually, I think we should not involve ourselves with exact meaning for species as a valuable term. As you know, we have two different concept in such cases: 

BSC: Biological Species Concepts

PSC: Photogenical Spspeciespts

Also, bootstrap meaning is more important than defines for species, I guess.

There are at least 27 species concepts, and in the last 70 years there has been much debate among biologist and philosophers on what defines species. I tried to shed some light on this case and the more modern additions to the debate in a paper in the Nordic Journal of Botany: https://www.researchgate.net/publication/255708030_Freakish_patterns_-_species_and_species_concepts_in_apomicts

Article Freakish patterns - species and species concepts in apomicts

Oh, and you can take a look here, in a discussion some weeks ago: https://www.researchgate.net/post/what_is_the_best_description_for_subspecies_and_which_factors_can_effect_to_produce_agroup_as_subspeices

This situation you described is very common for a number of plants for example.

There are several speciec concepts have already exists and all depends on what concept you will follow (biological, cladistic, phylogenetic, phenetic, eclolgical etc., the number of all concepts is near 26 or above). Species are logical categories, but individuals and populations are really existing objects.

Such facts you provide looks like speciation in action, where species aren't fully isolated. In these cases we can observe a number of introgressive hybridization.

Some literature about this problem http://www.ucl.ac.uk/taxome/jim/Sp/species.pdf

The question isn't "What defines a species" so much as who defines a species =)

It is difficult to define or to have an only concept, but when you have diferent evidences, taxonomic, phylogenetic with molecular data and another other source of evidence, you can be more confident when proposing a new species

This is especially according to your answer of which kind of species do you ask about? There are many kind of species and characteristics some of them have visual differences so that  you can easily think that they are different. But this is not my area so I cannot give a satisfactory answer for the definition of eukaryotic and multicellular organisms species. According to my experiences which I currently doing my PhD studying with halophilic bacteria from salty environments and previously with extreme halophilic archaea, the identification criteria of a species depends on their phenotypic, chemotaxonomic and genotypic properties and DNA-DNA hybridization. Especially there are many criteria that make a strain different from others in the same species and in the same genus that come across with geographic separations, lateral gene transfers, mutations inside the genomes, conjugations, effects of environmental stress conditions such as salt, U.V. high pressure, low temperature, high temperature and so on. Of course they have conserved gene regions which tells us that all species share a common ancestor and to know this is very exciting which shows us the concept of the evolution. So our aim is to determine the differencies between our isolates from the type strains. In genetically we often use 16S rRNA gene region comparisons for prokaryotes but as to me it only gives in genus-level identification mostly because of the conserved properties are not separate them for a very closer species of the same genus, same species. So the phenotypic and biochemical tests helps to find the answer. Sometimes the differencies in the phenotypic properties can separate two similar 16S rRNA gene region containing isolates from each other. İf there is a differencies in that phenotypic properties this will help you think that they can be different from each other. But the best is to use genome analysis or multiple sequence analysis to differentiate both of them as well. According to my opinion species will be different all the times from each other because of the environmental and evolutionary impacts, HGT, viral effects, transposons and so on. But also definite species has definite overall characteristics that define them as a species as the any phenotypic, biochemical or genetic property that is different from others, makes it a different strain and in course of time when the characteristics are much more differentiated from each other will make them different species in the same genus.

Hi, Species have many concepts: taxonomic speces, biology species, cladistic species, agamo species and ...

which of them is your view?

 If we can define a species in a simple way then we can say that these are the group of organisms which can interbreed each other and produce a fertile offspring. But this defination is not fitted to all cases. However, species are  grouped based on their physical  and genetic characteristics and placed under a genus because of their same ancestors. According to your question, we can say that in biology there are always exceptional/changing theories rather than exact. In many cases, there is change in DNA level of same species just because of Environmental factors. Even in some cases mutation can change the sequence of same interbreeding species. These are just the evolutionary changes in a species  to cope up with the changing environment. But firstly our main aim is to classification of an organism for the study. so we have to placed a particular species under a genus so that in future we can easily study them. 

There is enough theoretical work on species delimitation. Personally I like the phylogenetic species concept. In the practice, however, many taxa are defined at the taxonomist's discretion.