Brachiopods, sharks, crocodiles, coelacanth, ginko, horsetails and velvet works are all considered Living Fossils. We'll talk about those species and others on today's cast. Living Fossils are an interesting area in palaeontology. 
Today's episode was spurred by Sayre Morgan who asks the question, "if brachiopods are technically still alive today and look similar. And we can recognize them in the fossil record over 500 million years ago. Why do they maintain their original appearance?" 
There are many organisms we refer to as "Living Fossils." We have a perception — sometimes true and sometimes not — that there has been relatively little evolution in their body plan and style over time. Sometimes habitats — whether terrestrial or marine — and lifestyles remained so stable that there was little evolutionary pressure to change. There are still some evolutionary changes. For horsetails, we see a stable body design but huge changes in size. For our crocodylian friends, we often refer to them as "unchanged since the age of Dinosaurs" but that is not strictly true. They were massive back in the Triassic and slowly evolved and shrunk over time to match their environments. Similar stories are true for sharks, lice, coelacanth, velvet worms, water striders, lungfish... the list goes on. 
An excellent question, Sayre. I hope that helps!