This is a very nice piece of work but not exactly an atlas of ancient history. It is a cultural geography of Europe, North Africa, and the Middle East. It has far fewer place names than I wanted from an atlas and much of it is pre-history beginning some 40,000 years ago.
It starts with a brisk and entertaining account of the author's methods for interpreting scanty archaeological and linguistic evidence. This is at once accessible, learned, detailed, acerbic, and engaging. There is a funny bit about how archaeologists will say "a major new civilization" when they mean "a particularly disappointing dig", or will say "earliest known" when they mean "undated", and more. There is a terrific account of what it took to re-settle humans in Europe as the last ice age retreated. The effects were strong on Northern Europe into historical times (indeed Scandinavia and some of Russia is still rising and drying out today). The author estimates the human population of all of Europe and the Middle East was only about 100,000 in 9,000 BC.
The book describes movements of peoples, languages, technologies, and writing systems. It maps out the earliest known trade relations. It includes many maps but with few place names. Rather they indicate where various ethnic groups lived and what technologies were used where. As it enters historical times the book describes the campaigns of rulers and empires. It is a beautiful piece of work and beautifully concise.
On the other hand, if you are reading Euripides and you want to know where Lemnos was, you won't find it here. You will find the most famous places: In Greece, besides
Athens and Sparta, are Mycenea, Lesbos, Argos. That is like finding Chicago and San Francisco in an atlas of the US. But you will not find general Meno's birthplace of Larissa--which you would read about in either Plato or Xenophon. It is like not finding St. Louis in a US atlas. Those places are found in the Atlas of the Greek World (Cultural Atlas of) by Peter Levi. And they are found in another book you should read anyway, namely the Landmark Thucydides. It is a terrific edition available in paperback and it shows these places in detailed maps.