If I was a classy and hard-working diarist I'd find and embed an amazingly appropriate clip from the 1947 film noir after which I've named this diary. But I'm not and my discussion of 'living fossils' is far more appropriately introduced by this other fine piece of cinema. Check it out and then follow me after the fold (if you dare)
A fairly common theme in science fiction and fantasy is ancient buried organisms coming back to life and wreaking havoc. In real life we have lots of 'living fossils' and they tend to be more well behaved (my younger self would have said 'more boring'). Let's look at a few examples.
The classic living fossil is the coelacanth(genus Latimeria). This fish has most of the characteristics you would want in such an organism: it is big, weird-looking, lives in a remote and exotic location, and was discovered in an African fish market (where all the best living fossils hang out). Unfortunately its credentials are somewhat marred by never actually devouring anyone or menacing a major metropolitan area. However it does play a minor role in cinema history as the blood of a coelacanth is the putative cause of the 'throwbacks' in the film referenced at the start of the diary.

Coelacanth - Wikimedia Commons
More to the (biological) point, the coelacanth is an excellent candidate for living fossildom for several other reasons: it is a member of a group with an extensive fossil record extending from the Paleozoic through the Cretaceous, it has no living relatives (see below), and it is remarkably similar to many fossil species. The original coelacanth discovery was made in the 1930s when a specimen was found in a fish market. Working with local fishermen, the deep sea home of the coelacanth was found near the Comoros Islands in the Indian Ocean. For many decades specimens were only observed after death or while dying (changes in pressure and temperature meant that they did not live long once brought to the surface). Submersible observation in recent years has revealed that the coelacanths spend much of their time sheltering in caves or under overhangs. Also that there appears to be very few of them. In the 1990s a second very similar species of coelacanth was discovered off of Indonesia.
Coelacanths are one of three living lineages of a group called the Sarcoperygii or lobe finned fishes. The other two groups are the lungfishes and the tetrapods (i.e. amphibians, reptiles, birds, and mammals). Unlike other fishes the bones in their appendages are arranged in a row rather than the fan-like arrangement of the fins of a sunfish.
So, other than being the inspiration for horror movies, what can we make of living fossils? First, not all of them have the drama of the coelacanth. The tree, Gingko biloba, is a common urban tree in temperate North America. It is perhaps best known in areas where it is common for the foul smelling 'fruit' produced by female trees. Although not as ancient as the coelacanth, Gingko was common type of tree during the Mesozoic. It's distinctive fan-veined leaves can be seen commonly in the fossil record. The one living species is from the far east where it is commonly cultivated in temple gardens. Wild populations are known from China but it is unclear if they are derived from temple stock. It may be that no wild populations of this plant remain.

Wikimedia Commons
There are any number of other examples. Horseshoe Crabs (actually more closely related to spiders than to crustaceans) which have only 4 species worldwide occurring geographically isolated coastal areas (one of which is eastern North America). Brachiopods, a phylum of clamlike animals, were a dominant form of marine life in the Paleozoic and now consist of a small number of mostly deep sea forms. Similar crinoids (stalked relatives of starfish) are abundant fossils and are now a fairly obscure group although they are found in both the deep sea and coastal areas. Cycads (palmlike plants) were the dominant form of large vegetation in the early and middle Mesozoic. Today a couple of hundred species remain but almost all of them are restricted to extremely small geographic areas. The 'Sago Palm' a common ornamental plant in the south is a cycad. Another vertebrate example are the two Tuatara species, lizard-like reptiles from New Zealand. They are the only surviving examples of a large group of fossil species. A particularly charismatic living fossil is the chambered nautilus, a close relatively of octopus and squid which has retained the shell.

Horseshoe crabs (Florida)

Crinoid (Great Barrier Reef)


Cycads in the Miami and Sydney Botanical Gardens showing conelike reproductive structures.

Semi-wild Tuatara in Wellington New Zealand

Chambered Nautilus - Wikimedia Commons
Technically these are called relictual species. Typically they occur in highly isolated parts of the world. The deep sea seems to be a common location for marine forms. Areas that are geologically ancient but cut off in some way, either by the ocean (New Zealand) or climate (South Africa or the moister regions of Australia which are home to many cycads). In these areas members of ancient groups have been able to avoid extinction.
But not to avoid evolution. One reason I wrote this diary is to address a common misconception. Evolution doesn't have end points, other than the final one of extinction. If a population exists it is evolving. If nothing else, noncoding DNA experiences mutation and genetic drift and the sequences slowly change over times. For a species to stay the same, natural selection has to keep it that way. This is known as stabilizing selection. The 'living fossils' are still here and in the forms we see because natural selection has maintained those forms over millions of years. Within their limited world they are great evolutionary successes.
I'd like to wrap up by pointing out some variations on this theme. Lampreys and hagfish are jawless vertebrates. Their mouths have rasping teeth but no jaws. They have a number of characteristics that indicate that they each (separately) branched off to form their own lineages prior to the evolution of jaws. Because they retain a number of these ancestral characteristics you might be tempted to call them primitive. However they have been evolving for as long as us - just in different ways. Both groups are highly specialized: hagfish are scavengers and lampreys are parasites. Their body form is highly different from that of fossil jawless fish. They have evolved ways of life to take advantage of the jawless condition.
The Onycophora are a group of animals known as velvet worms. They are thought to be the closest living relatives of the arthropods. They look something like a cross between a caterpillar and a centipede. They have an ancient fossil record, dating back to the start of the Paleozoic is not before. The fossils are marine however and all the living species occur in tropical forests. So although they maintain the basic body form they have obviously evolved in other ways.

Onycophoran - Wikimedia Commons
In contrast we have a number of ancient groups that are still on earth today and have not changed radically in appearance. These include turtles, sharks, dragonflies, etc. Although most of these groups have fairly small numbers of species compared to close relatives they are still abundant, conspicuous, and ecologically important. I think they are just a valid living fossils as the coelacanth although few people would think of them that way.

Dragonfly - Florida