The Eocene epoch, from 55.8 to 33.9 million years ago, began about 10 million years after the extinction of the dinosaurs, when mammals were diversifying and had already risen to some prominence, occupying most available niches. As was the case throughout most of the Earth's history, climate during the Eocene was relatively balmy, with tropical conditions extending up to 45 degrees from the equator, and a temperate climate extending to the poles. During the Eocene, the climate at the poles would have been comparable to that of the Pacific Northwest.
The continental arrangement during the Eocene was similar to that of today's, except Antarctica was still connected to South America and large parts of Europe, the Middle East, and Southeast Asia were flooded. This is due to the near-absence of continental ice caps at the poles, meaning all that ice was in water form, filling up the oceans. Because Antarctica was still connected to South America, there was no frigid circumpolar current. Antarctica was temperate, and the worldwide circulation of ocean currents gave the global climate a degree of homogeneity that has not been seen since.
The Eocene was somewhat before the Age of Grasses, meaning that most of the planet was covered in forests rather than grasslands. Arboreal mammals were the most common, and almost all animals were small -- few were over 10 kg (22 lb) in size. On average, they were 60% smaller than the Paleocene animals that came before them, and even smaller relative to the large animals that would evolve shortly after the conclusion of the epoch. It is thought that these small sizes helped animals cope with the heat better.
The representatives of many modern mammalian orders are thought to have originated during the Eocene, including bats, proboscidians (elephants and relatives), primates (though these may have evolved much earlier), rodents, and many marsupial groups. "Modern" plant and animal life evolved during this time, meaning it would have had many of the earliest ecosystems that people today would have viewed as normal rather than distinctively ancient, like the cycad/dinosaur ecosystems of the Mesozoic.