The Summa Theologica is the magnum opus of St. Thomas Aquinas, a 13th century theologian and a seminal work. It is, perhaps, the defining document for medieval theology. Although not principally an apologetic work, the summa does address the question of god's existence as Question 2 of its first part. Specifically, in Article 3 Aquinas raises five "proofs" (or "Ways") of the existence of god:
The "First Way" is the Cosmological Argument from an Unmoved Mover (change), followed by a Cosmological Argument from a First Cause (cause).
The third so-called proof is an Argument from Contingency.
The fourth is an Ontological Argument from perfection. Note that Aquinas rejects Anselm's version of the argument in Article 1, Reply to Objection 2.
His fifth so-called proof is a Teleological Argument, an argument from design (specifically, from the design apparent in the ways inanimate things behave, according to Aquinas). Aquinas lived before Darwin and did not understand how Natural selection can cause complexity in life. Aquinas also apparently believed that inanimate objects have a purpose, for example that the sun and moon exist in order to give light. Water moves downhill in order to join streams, rivers and the sea -or to evaporate in a dust bowl.
It should be noted that Aquinas also attempts to refute the Argument from Evil (quoting Augustine), and the Argument from Simplicity.
Thanks to the modern wonder of the Internet, and the generosity and kindness of newadvent.org, there is easily searchable and readable English version of the Summa Theologica online .