Leading proponents of reliablist theories of knowledge and justification have included Alvin Goldman, Marshall Swain, and more recently, Alvin Plantinga. Goldman's article "A Causal Theory of Knowing" (Journal of Philosophy, v. 64 (1967), pp. 335-372) is generally credited as being the seminal statement of the theory, though D. M. Armstrong is also regarded as an important source of the theory.
On the classical or traditional analysis of 'knowledge', one must be justified in believing that p in order for that belief to constitute knowledge; the traditional analysis has it that knowledge is no more than justified true belief (see justified true belief analysis). Reliabilist theories of knowledge are sometimes presented as an alternative to that theory: rather than justification, all that is required is that the belief be the product of a reliable process. But reliabilism need not be regarded as an alternative, but instead as a further explication of the traditional analysis. On this view, those who offer reliabilist theories of justification further analyze the 'justification' part of the traditional analysis of 'knowledge' in terms of reliable processes. Not all reliabilists agree with such accounts of justification, but some do.
Some find reliabilism objectionable because they believe it entails externalism, which is the view that one can have knowledge, or have a justified belief, despite not knowing (having "access" to) the evidence, or other circumstances, that make the belief justified. Most reliabilists maintain that a belief can be justified, or can constitute knowledge, even if the believer does not know about or understand the process that makes the belief reliable. In defending this view, reliabilists (and externalists generally) are apt to point to examples from simple acts of perception: if one sees a bird in the tree outside their window and thereby gains the belief that there is a bird in that tree, they might not at all understand the cognitive processes that explain for their successful act of perception; nevertheless, it is the fact that the processes worked reliably that accounts for why their belief is justified. In short, they find theirself with a belief about the bird, and that belief is justified if any is, but they am not acquainted at all with the processes that led to the belief and made them justified in having it. Of course, internalists do not let the debate rest there; see externalism (epistemology).
Another of the most common objections to reliabilism, made first to Goldman's knowledge reliable process theory of knowledge and later to other reliabilist theories, is the so-called generality problem, as follows. For any given justified belief (or instance of knowledge), one can easily identify many different (concurrently operating) "processes" from which the belief results. My belief that there is a bird in the tree outside my window might be accorded a result of the process of forming beliefs on the basis of sense-perception, of visual sense-perception, of visual sense-perception through opaque surfaces in daylight, and so forth, down to a variety of different very specifically-described processes. Some of these processes might be statistically reliable, while others might not. It would no doubt be better to say, in any case, that we are choosing not which process to say resulted in the belief, but instead how to describe the process, out of the many different levels of generality on which it can be accurately described.
Many reliabilists also subscribe to the causal theory of belief.
This article needs a good meaty discussion of the distinction between reliable process theories and reliable indicator theories--Swain being a leading proponent of the latter.
Richard Foley has offered an internalist refutation of reliabilism in his article "What's Wrong with Reliabilism?"