The Quinque viæ (Latin for "Five Ways") (sometimes called "five proofs") are five logical arguments for the existence of God summarized by the 13th-century Catholic philosopher and theologian Thomas Aquinas in his book Summa Theologica. They are:
the argument from "first mover"; the argument from universal causation; the argument from contingency; the argument from degree; the argument from final cause or ends ("teleological argument").
Aquinas expands the first of these – God as the "unmoved mover" – in his Summa Contra Gentiles.
Background
Need for demonstration of the existence of God
Aquinas thought the finite human mind could not know what God is directly, therefore God's existence is not self-evident to us, although it is self-evident in itself. On the other hand, he also rejected the idea that God's existence cannot be demonstrated: although it is impossible to give a so-called demonstration, going from the causes to the effects; still, the proposition God exists can be "demonstrated" from God's effects, which are more known to us, through a so-called demonstration. However, Aquinas did not hold that what could be demonstrated philosophically (i.e. as general revelation) would necessarily provide any of the vital details revealed in Christ and through the church (i.e. as special revelation), quite the reverse. For example, while he would allow that "in all creatures there is found the trace of the Trinity", yet "a trace shows that someone has passed by but not actually who it is."
Categorization
The first three ways are generally considered to be cosmological arguments. Aquinas omitted various arguments he believed to be insufficient or unsuited, such as the ontological argument made by Anselm of Canterbury.
Source
A summary version of the Five Ways is given in the Summa theologiae. The Summa uses the form of scholastic disputation (i.e. a literary form based on a lecturing method: a question is raised, then the most serious objections are summarized, then a correct answer is provided in that context, then the objections are answered).
A subsequent, more detailed, treatment of the Five Ways can be found in the Summa contra gentiles. Aquinas further elaborated each of the Five Ways in more detail in passing in multiple books.
Essential and accidental causal chains
The first two Ways relate to causation. When Aquinas argues that a causal chain cannot be infinitely long, he does not have in mind a chain where each element is a prior event that causes the next event; in other words, he is not arguing for a first event in a sequence. Rather, his argument is that a chain of concurrent or simultaneous effects must be rooted ultimately in a cause capable of generating these effects, and hence for a cause that is first in the hierarchical sense, not the temporal sense.
Aquinas follows the distinction found in Aristotle's Physics 8.5, and developed by Simplicius, Maimonides, and Avicenna that a causal chain may be either accidental (Socrates' father caused Socrates, Socrates' grandfather caused Socrates' father, but Socrates' grandfather only accidentally caused Socrates) or essential (a stick is moving a stone, because a hand is simultaneously moving the stick, and thus transitively the hand is moving the stone.)
His thinking here relies on what would later be labelled "essentially ordered causal series" by John Duns Scotus. (In Duns Scotus, it is a causal series in which the immediately observable elements are not capable of generating the effect in question, and a cause capable of doing so is inferred at the far end of the chain. Ordinatio I.2.43)
This is also why Aquinas rejected that reason can prove the universe must have had a beginning in time; for all he knows and can demonstrate the universe could have been 'created from eternity' by the eternal God. He accepts the biblical doctrine of creation as a truth of faith, not reason.
For a discussion of a causal chain argument that is based on a created beginning, see Kalam cosmological argument.
The Five Ways
First way: The Argument of the Unmoved Mover
Summary
In the world, we can see that at least some things are changing. Whatever is changing is being changed by something else. If that by which it is changing is itself changed, then it too is being changed by something else. But this chain cannot be infinitely long, so there must be something that causes change without itself changing. This everyone understands to be God.