You can prove that God--in the traditional theistic sense--does exist, but you cannot prove that He does not exist for the simple reason that without the concept of God, proof is not logically possible for anything.
First, to prove He exists, which can be easily done by thinking about chains of essentially ordered efficient causation. Think about anything in existence right now--for instance, my finger pressing on a keyboard at a computer. For my finger to be pressing the keyboard, the motor neurons and stuff must be firing a certain way, and if they stop firing that way then my hand will stop pressing. So, my finger pressing in the way it does depends upon my neurons firing in a certain way. Then, my neurons firing in a certain way depends upon the atoms and such involved in the neurons firing to be acting a certain way. Then, the atoms acting in a certain way are dependent upon the quarks making them up acting in a certain way, etc. Every step in this chain of essentially ordered causes depends upon something more fundamental and necessarily existent than it; furthermore, if for some reason the motor neurons were to stop firing, my finger would stop pressing, and if the atoms would stop acting in a certain way, the neurons would stop firing, and so on and on. The skinny is that, for anything at all to be happening in the universe, or for anything to exist, there must be a most fundamentally existent layer to this essentially ordered causal chain, and this layer--or, more appropriately, thing--is what we mean by God. Therefore, God exists.
This is only one logically certain and indubitable proof of the existence of God, which has never been refuted satisfactorily. Many others are possible.
On the other hand, it is impossible to prove God doesn't exist, since proving something requires standards of truth and falsity. Yet, the standard of truth and falsity would necessarily be, in fact, the most fundamentally existent thing which I described above, since truth and falsity are actually just names for "what is" (aka truth) and "what is not" (aka falsity). Thus, without God (aka most fundamental and necessary Being), there is no way that proof or disproof is possible. Hence, you can prove God exists without a shadow of a doubt, but cannot prove he does not exist since doing so would require you to forfeit the possibility of proof itself.
You don't have to look very far to see that God, Santa, and Bigfoot are all related. This is because they are all made up. It is proven in the fact that their are hundreds of different religions. In order for anyone of them to be right the others would all have to be fake. If hundreds are made up, they all are.
In my opinion God exists but his existence would be very hard to prove. So I think the question of his existence is best disregarded or taken on faith. Many people believe that God exists and site such things as creation and the presence of intelligent life on earth as their "proof." Many people believe that God does not exist and site such things as the vast cruelty and inequalities that exist in our world as their "proofs." Devising a valid test for the existence of God is a task that has defied philosophers, clerics, sages, and the common man forever. Perhaps such a test could be discovered. But if that is true, why has it not happened by now? Methinks the more important place for each of us to focus attention, rather than on proving or disproving God, is to concentrate on living a responsible, constructive life. That is activity that in my opinion is more apt to have benefited each of us when we look back on our lives at age 92 when facing our graves, than decades of logical or philosophical hair splitting. One might as well debate God's sex as God's existence.
God is considered to be a supernatural being, his power, presence and actions are all supernatural. While his existence is always in question, it can be accepted by all that regardless of the truth behind his existence...He is believed to be supernatural. The simple fact that (If he does exist) he is a supernatural being invalidates any and all logical arguments for OR against his existence. How do you use logic, reasoning, fact or evidence to prove or disprove the existence of something that cannot be explained? It cannot be done. Try proving/disproving the Marfa Lights...It cannot be done. The existence of God has never truly been argued properly because it cannot be logically debated by definition, you may argue proof of other theories that might support God and you may argue the invalidation of certain aspects of the world and universe you you cannot truly argue God for the same reason that you cannot argue for or against anything else that falls under the category of supernatural. Possibilities can be argued all day...But always serve as weak arguments.