It was Edmund Husserl who inaugurated the movement known as Phenomenology. He was closely concerned with issues pertaining to the foundations of mathematics and logic as well as numerous questions in psychology being discussed in his time. Phenomenology became a wide and varied field, incorporating many thinkers throughout the 20th century. It remains a major school of thought, though its influence has become more diffuse. Husserl believed that he was providing a form of philosophy that overcame the modern problem of Idealism, allowing philosophers once again to discuss the "things in themselves." Phenomenology became a study of the way that things "appear"—how they come into awareness and just how they are constituted by the human knower. He is known for works such as Logical Investigations, Ideas (in numerous versions), Cartesian Meditations, and Formal and Transcendental Logic.

Perhaps you do not know all of these thinkers, though the names are likely to be somewhat known to you, at least from lists in texts. Sadly, America has not existed long enough to create a large group of philosophers as was the case in ancient Greece, the High Middle Ages (or even the so-called period of "Silver Scholasticism" in Spain and the Low Countries), or modern Europe. Still, there have been some, but among their number has never been Bertrand Russell. Lord Russell is an Englishman—a great logician and mathematician but not an American philosopher.

The 20th century political philosopher, Leo Strauss, believed that many political and philosophical writings contain esoteric meanings. He came to this conclusion based on his interpretations of the writings of the Muslim philosopher al-Farabi and the dialogues of Plato. Plato's dialogues are particularly well known for the use of many layers of irony, wit, and story to express deep philosophical truths. Strauss (and his followers, who are called "Straussians") came to use this method for interpreting many texts.

He was very influential, given his time as a professor at the University of Chicago, and is often associated with the American political movement known as neoconservatism. This association is not quite true, however. Really, if his followers share anything in common, it is a devotion to a close reading of texts, often those of Greek thinkers but also including others in the canon of philosophy and political philosophy. Their readings do tend to be so close as to look for many such hidden shades of esoteric meanings.

Kierkegaard, Dostoevsky, and Nietzsche, although all nineteenth-century Europeans, were actually quite different thinkers, but all were exemplars of Existentialism because they focused on the individual human and the sense of alienation the individual felt in modern society. In the early twentieth century, authors and philosophers picked up on the common themes these authors highlighted.

Both Friedrich Engels and Karl Marx were German-born anti-capitalists active in London's expatriate radical community. In 1848, they wrote and published The Communist Manifesto as part of a communist organization, originally in German. Their critique of capitalism from a Hegelian viewpoint crystallized reaction to capitalism, and launched the most influential political movement of the twentieth century.

Henry David Thoreau was a nineteenth-century author and philosopher from Massachusetts who was one of the leading figures of the transcendentalism movement. Transcendentalism focused on the goodness of nature, human possibility, and a free-form spirituality that separated itself from Christianity. Thoreau's Walden, a meditation on the two years he spent in a cabin in the woods by Walden Pond, is considered a classic of the genre.

The philosopher Friedrich Nietzsche was an almost revolutionary figure in philosophy during the late nineteenth century. Trained as a scholar of classical philosophy, most of Nietzsche's works, like Thus Spake Zarathustra, The Twilight of the Idols, and Beyond Good and Evil, challenged the philosophical norms of his time. Nietzsche argued against strict rationality, traditional morals, and the view of humanity during the nineteenth century.

Soren Kierkegaard worked in relative obscurity for a giant of Western philosophy, writing only in Danish during the early nineteenth century without a University posting. Although not widely read in his  lifetime, Kierkegaard became hugely influential in the twentieth century and is considered one of the intellectual founders of existentialism. Kierkegaard's focus on individual thought, the subjective nature of human reality, and a critique of traditional religion were all forerunners of twentieth century existential thought.

The work Utilitarianism (1863) was written by John Stuart Mill. He was a prodigiously educated man, taught in a thorough manner by his father James Mill and influenced heavily by Jeremy Bentham. This hard education in his youth actually led to something of a mental breakdown, from which he recovered relatively quickly. The work Utilitarianism is a kind of extension or continuation of Bentham's thought on moral philosophy. It is best known for the principle that states, simply speaking, that the best action is the one that brings about the greatest amount of happiness.

This outlook can become very subjective depending on how one understands it. There remain many philosophers who are utilitarian in bent to this very day, so Mill's work retains great influence in almost all courses on ethics taught in undergraduate institutions.

Hegel is perhaps most famous for having written his dense tome The Phenomenology of Spirit (1807). Ostensibly, this work traces the different forms that human consciousness can take, working through pure sense perception, then to various kinds of scientific, moral, aesthetic, and historical modes of consciousness. The work is, however, an incredibly difficult text to read. The prose is dense and it is often difficult to distinguish between the theoretical and historical claims that Hegel is making in the course of his argumentation.

For all of that difficulty, however, the work was very influential, spawning numerous different schools of interpretation. Although he influenced most thinkers coming after him in continental Europe, his most famous "follower" was Karl Marx. Marx really cannot be considered a direct disciple of Hegel, for he significantly altered his viewpoint. However, he took over the notion of the dialectic of reason exposited by Hegel, using it in his own materialistic philosophy.