If from continental philosophy you throw out transcendental phenomenology and older idealist trappings–transcendental subjects and so on–you are left with a system which still has two components: the world and the self. It was the relationship between these two that took hold as the major problem for 20th C. continental philosophy.
The upshot of the first phase of the “analysis of the self” we know as existentialism and may be traced back to Kierkegaard, Nietzsche and others, but it really gets off the ground as an independent topic with Heidegger. His pre-conscious self, which he calls Dasein, is primarily characterized by two things: 1) it is ontologizing (it takes part in giving the world its particular character); 2) it cares.
This second essential component of the self, that it cares, is, incidentally, the basis on which Hubert Dreyfus (a Heidegger scholar at MIT) in the 50s predicted the failure of strong AI. This prediction was, at the time, the basis of his being ostracized at MIT–his skepticism was unwelcome because significant amounts of funding and enthusiasm lay behind the idea that real AI was just around the corner. The caring, or concerning, nature of the self appears today translated to the framing problem within AI research–that is, how do you tell a robot that it should care, without telling it what it should care about? How does one “take in the whole world” and yet at the same time have an appropriate response to it? Human beings come to the world with concerns that help us shape it and relate to it; this is the core of what it means to Heidegger to have a self .
Subsequent philosophers, still with the same problem of the relationship between the self and the world, have turned to an alternative vein of thought running in parallel through continental philosophy: that of Marx, Freud, and Nietzsche in the 19th Century and Lacan, Foucault and Derrida in the 20th.
These philosophers are all materialists, and in their materialism they were able to conceive of a different kind of human subject (self) that idealists who were still preoccupied with a transcendental subject had trouble thinking about: a subject that is essentially divided; that has an unconscious; whose desires and motivations are “always already” suspect; whose self-knowledge is a game of cover-up, confusion and mistake; that can be more easily known by others than by it itself.
(One needn't subscribe to materialism to endorse this picture of the self, of course, and indeed much of French philosophy from the 60s onward is an attempt to phenomenologize this project.)
The 19th C. thinkers I've mentioned introduced into our culture a revolution in the way we think, and profoundly changed how we think of ourselves today. They introduced deception and doubt at the heart of the self itself. No longer could one be religious based on the conviction that one had clearly identified god “at work in the world.” The obviousness of ideas themselves was a testament to their ideological and suspicious origin. How could one seriously remain a Christian–or the same kind of Christian–when Nietzsche identified resentment at the heart of Christian thought? How seriously could one now say, as Thomas Aquinas had, that God is so Just and Good as to let all those in heaven see the suffering of those in hell? How seriously could one claim that one’s prejudices were natural, that one’s position in the world was just, when Marx's Theory of Ideology accounted for all these self-serving rationalizations and more? How could one seriously claim to believe with sincerity anything, when underneath it all were our basic, blind libidinal drives?
This assault on the integrity of the self goes on today. Slavoj Zizek, for example, brings all these thoughts and more together, launching blistering assaults and subversive critiques and interpretations of the world and ourselves. For Zizek, action itself is inherently suspect; you cannot just go around protesting things, or doing things, because the very conditions under which you dream of a better future are often still fully internal to the system of oppression you are currently fighting. That is, the emancipatory gestures that you believe yourself to be taking are often the very acts which preserve the system of oppression that you are caught up in.
And so the problematic nature of the self and its relationship to the world continues to pose new and difficult problems.