With all the science of genetics, and genetic engineering, would reviving an extinct species possible?
Yes, akin the story of "Jurassic Park".
Would it be possible now, with the science we have today?
This is a slightly tricky question - in that it's certainly plausable that some species could be resurrected, but probably not others (at least, not with technology as it stands).
The easiest way - cloning from preserved tissue/cells.
For recently extinct species that have close living relatives, cloning is one way get them back. You could take the nucleus from a normal 'body' cell in the preserved tissue and implant it in the egg or a modern relative of the original species (so called Somatic Cell Nuclear Transfer). Researchers tried this with a sub-species of goat that went extinct at the turn of the century, but failed. It's likely that part of the problem here, is that embryonic development of mammals depends greatly on cues from the mother - and the domestic goat used in these attempts may have been a poor surrogate. I gather the mammoth example cited in another answer is attempting to take the same route
A harder way - creating a synthetic genome
For most species that people would want to resurrect, well preserved cells or tissue just aren't avaliable. In these cases it might be possible to use ancient DNA techniques get genomic sequences (especially creatures that die in permafrost like mammoths!). The problem here is going from DNA sequences as letters on a computer to a physical genome. It's possible to make a synthetic bacterial genome, but even that had to be inserted into an existing cell. Inserting a eukarotic genome into the nucleus of egg cell would be much harder, because eukaryote genomes are (mainly) big, and packaging them into chromosomes is very complex (try to ignore the 90s webdesign there...).
Once you'd done that you still have the problem of finding a surrogate the develop the eggs, or a machine to do the job. Not easy.
It's possible we will eventually be able to do all this. But it's a long from the synthetic bacterial genome to there.
If you count viruses, then yes, it has been done: Reconstruction of the 1918 Spanish flu pandemic virus
All the information that you need to revive an extinct specifies should be stored in the DNA. With present technology that means that you can theoretically revive species that died in the last ~100 000 years. There are currently efforts underway to revive mammoths.