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from Thomas Aquinas (1225-1274),  Summa Theologica

Part One, Question II deals with "The Existence of God."  The first article asks "Whether the existence of God is self-evident."  (Aquinas argues that it is not.)  The second article asks "Whether it can be demonstrated that God exists."  (Aquinas argues that it can.)  The following excerpts come from the third article, "Whether God exists."

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Objection 1.  It seems that God does not exist, because if one of two contraries be infinite, the other would be altogether destroyed.  But the name GOD means that He is infinite goodness.  If, therefore, God existed, there would be no evil discoverable; but there is evil in the world.  Therefore God does not exist. 
 . . . 

I answer that, The existence of God can be proven in five ways.

 The first and most manifest way is the argument from motion.  It is certain, and evident to our senses, that in the world some things are in motion.  Now whatever is moved, is moved by another, for nothing can be moved except it is in potentiality to that toward which it is moved; whereas a thing moves inasmuch as it is in act.  For motion is nothing else than the reduction of something from potentiality to actuality.  But nothing can be reduced from potentiality to actuality, except by something in a state of actuality.
 .  .  .  It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e., that it should move itself.  Therefore whatever is moved must be moved by another.  If that by which it is moved be itself moved, then this also must be moved by another, and that by another again.  But this cannot go on to infinity, because there would be then no first mover, and, consequently, no other mover, seing that subsequent movers move only inasmuch as they are moved by the first mover; as the staff moves only because it is moved by the hand.  Therefore it is necessary to arrive at a first mover, moved by no other; and this everyone understands to be God. 

 The second way is from the nature of efficient cause.  .  .  . There is no case known (neither is it, indeed, possible, in which a thing is found to be the efficient cause of itself..  .  .   Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God. 

 The third way is taken from possibility and necessity .  .  .  If at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence--which is absurd.  .  .  .   Therefore we cannot but admit the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity.  This all men speak of as God. 

A fourth way is taken from the gradation to be found in things.  Among things there are some more and some less good, true, noble, and the like.  But more and less are predicated of different things according as they resemble in their different ways somethings which is the maximum, as a thing is said to be hotter according as it more nearly resembles that which is hottest.  .  .  .  Therefore there must be something which is to all beings the cause of their being, goodness, and every other perfection; and this we call God.  

 The fifth way is taken from the governance of the world  .  .  .  We see that things which lack knowledge, such as natural bodies, act for an end, and this is evident from their acting always, in the same way, so as to obtain the best result.  Hence it is plain that they achieve their end not fortuitously, but designedly.  Now whatever lacks knowledge cannot move towards an end, unless it be directed by some being endowed with knowledge and intelligence, as the arrow is directed by the archer.  Therefore some intelligent being exists by whom all natural things are directed to their end; and this being we call God. 

 Reply Obj. 1.  As Augustine says: "Since God is the highest good, He would not allow any evil to exist in His works, unless His omnipotence and goodness were such as to bring good even out of evil."  This is part of the infinite goodness of God, that He should allow evil to exist, and out of it to produce good.