Cosmological arguments were originally introduced to Islam in the 9th century by Avicenna, and to Christianity in the 13th century by Thomas Aquinas; both based on Platonic-Aristotelian ideas about causation. Supporters of this strand of argument invoke some aspect of the seemingly unceasing event stream in the Universe to try to show that a God lies at the beginning of everything.
There have been many variations of the cosmological argument. Plato and Aristotle believed reality was eternal, but included a privileged being (many in the case of Aristotle) to set the physical world in the state of motion it's currently in. Thomas Aquinas himself presented two variants of the argument in his "5 Ways"; one concerning motion, and a similar one for causation. An equivalent line of reasoning concerning "formal causes" (as opposed to efficient ones) is associated to Leibniz' work on logic. The "Kalam" cosmological argument from medieval Islamic philosophy has found new vigour among contemporary theologians. Every event in the Universe is caused by something else, as opposed to existing by logical necessity or accident. Acceptance of this premise leads to one of three options: either the causal chain extends infinitely in the past, history is a finite loop that repeats itself over and over, or a supernatural (i.e. "not caused") first cause marks the beginning of everything. From there it is posited that the first two options are to be ruled out for their absurdity or incompatibility with current scientific evidence (e.g. the age of the Universe is finite). Therefore, the cosmological argument concludes, nature comes from a supernatural cause. This cause is equated to God.
Philosopher David Hume objected to the very idea of causation as a necessary feature of the world. Experience of a causal chain is only apparent, since we never manage to logically establish the dependence of the present on the past. According to Hume, there is no logical contradiction in imagining an event for which no cause or effect is to be found. Immanuel Kant rejected the cosmological argument, as well as most of metaphysics, on the basis that it is not epistemologically sound to infer facts about reality "as is" (noumenal reality) from mere human experience (phenomenal reality). Knowledge is limited to tautological/analytic statements (logic, math) and synthetic a posteriori statements (empirical science), whereas a priori reasoning is insufficient to resolve synthetic statements. The cosmological argument rests on the supposed impossibility of an infinite regress. Modern set theory, and therefore mathematics, finds no contradiction in using such concepts. Bertrand Russell points out on his criticism of Thomas Aquinas that "[e]very mathematician knows that there is no such impossibility".
The following syllogism forms the core of the contemporary Kalam cosmological argument, as presented by William Lane Craig: Everything that begins to exist has a cause. The Universe began to exist. Therefore, the Universe has a cause.
Rejecting the premises
It has been pointed out that premise 1 commits a fallacy of composition. That is, assuming the whole (the Universe) must also display features which are reasonably attributed to the parts. Namely, the property of having a cause. The argument is made for an uncaused cause at best; however this "monadic" being need not look anything like a deity whose attributes are intelligence, purposefulness and everything else that might resemble a human-like God. In other words, cosmological arguments jump from a trivial metaphysical intuition to an unjustified anthropocentric, theological claim. As an example of alternatives to God which would also benefit from cosmological arguments, cosmologist Max Tegmark hypothesises that ultimate reality may be just a "Mathematical Universe" made of all logically necessary facts. Similarly, it doesn't follow from accepting the conclusion (the Universe has a cause) that this cause itself was not in turn caused by something else. According to cosmologist Sean Carroll, the premises are an oversimplification of our current state of knowledge in physics, and a misuse of the same thereafter. There is room for naturalistic explanations to the Big Bang. Strictly speaking, Big Bang theory doesn't claim the Universe began to exist in the 'ex nihilo' sense that is required by Abrahamic religions. Determining what caused it to expand from a very condensed state is a major open problem in cosmology.