The following are the most important arguments for and against theism (the existence of God). I do not endorse any of the below claims strongly. This page is for educational purposes.
Argument from Religious Experience.
Claim: God must exist because we have religious experiences which are unlike our experiences of the physical world.
Response: But religious experiences could be caused by neural firing or by demons.
Argument from Probability.
Claim: In all probability, given the evidence, and Bayes' Theorem, God likely exists. (Swinburne).
Response: But if you set the probability of this universe very low, then you need to set the probability of theism very low, in Bayes' equation, since a very high number in the numerator (e.g. that you think Prima Facie God exists), would result in a very low posterior probability (after calculation) if the universe Prima Facie would not exist (the denominator). If you follow the maths and then set the numerator low, you're effectively admitting the chances that God exists are low, Prima Facie.
Claim: The fact that this universe contains life is a miracle because it was so improbable that the fundamental constants that make life possible have a wide range of possible values, so, God must have set them to create life. (Lane Craig gives a good version of this in his exposition of the Islamic Kalaam argument).
Response: Stenger shows that a quite wide range of values (10^500) could have produced a life-bearing universe; moreover if there are millions of parallel universes, life certainly exists in one of them.
Argument from Providence.
Claim: The fact that our lives are mostly good and mostly pleasant with blessed experiences such as love, children, etc., that God must exist, because only a good God would allow such things.
Response: Benatar argues that life is mostly hard and therefore we should stop having children. This is because they will mostly suffer, and you're imposing suffering on new beings, which is unethical. This is a position called Antinatalism.
Claim: Our world seems designed for a purpose; to enable life. Just as if we find a watch in a desert, we would assume it was not there by accident but was created for a purpose. Life was intelligently designed. We do not know how life started.
Response: Our life-teeming world is more like a watch factory with parts available everywhere, than a desert. It's easy to create a watch in a watch factory. Also, a "purpose" is a question begging argument because it implies a person who has that purpose. Meaning you can only make sense of the "teleological" argument if you assume that there's a Person with a purpose (ie that God exists). Scientists admittedly have not solved the question of abiogenesis (how life started from non-life), but, there are several theories and papers on the topic recently which offer reasonable theories - more explanatory than "Then God said" (which is tantamount to saying it was just a miracle, not an explanation).
Argument from Scriptural Authority.
Claim: Scriptures are old and have stood the test of time, therefore they must be true and by inference, their contents must be true, including reference to God.
Response: But then any suitably old book must be totally true, even if it refutes itself or refutes other old books. The Gita is proof Rama exists, the Quran is proof that Allah exists, etc. Spiderman comics are proof that Spiderman exists, etc.
Argument from Consciousness.
Claim: We have consciousness which is amazing and unexplained by science. Therefore God created us and imbued us with a soul (consciousness).
Response: This is question-begging as it assumes consciousness cannot be explained, or will forever remain unexplained. It's a version of God of the Gaps. Over time we have seen our knowledge gaps, which we used to explain by reference to God, now have scientific explanations. So we assume that within a short time, we will have an explanation for consciousness.
Argument from Miracles.
Claim: Since miracles occur (e.g. when someone survives a plane crash or cancer), it follows that God exists.
Response: (a) That's survivor bias, (b) confirmation bias, (c) what about the other 199 passengers or cancer patients who didn't survive and (d) no one has regrown a limb no matter how much they prayed.
Argument from Human Morality.
Claim: Humans have moral behaviour. Only a good God would create us with moral behaviour. Therefore God exists.
Response: Apes and others display moral behaviour therefore it is a side-effect of group structures in gregarious or herd animals. Morality can be better explained by evolution, since non-gregarious animals display low moral scruples.
Evidential Argument from Evil.
Claim: Evil is allowed by God because he is testing our faith. Since he gave us free will to choose good or evil, it is up to us to choose or not choose evil.
Response: There is too much evil in this world, it is too widespread, and the individual acts of evil are too horrible, for them to be tolerated by a good God. Only a neutral or evil god would allow this. Furthermore, some cases of evil (e.g. the ability to suffer needlessly from accidents), supports the theory of evolution (that we evolved the ability to suffer to ensure that we avoid harm). Lastly, the biblical story is clear that the Devil gives free-will to Eve. Not God.
The Problem of Evil.
Claim: Evil exists. God is omnibenevolent (all good, perfectly good), therefore God should not create or allow evil. God is also omnipotent (all powerful), so he is perfectly able to stop any evil. He is also omniscient (knows all), therefore he knows about all evils. Therefore because God is God, he must stop all evil. Yet God does not stop evil. Therefore he does not exist or he is not all good, all knowing, or all powerful (ie he is not God).
Response: God allows only those evils that he could not stop without causing an even greater evil (Plantinga's "could not properly eliminate"). Also, there's theodicy of suffering: God allows evil because suffering builds character and makes us worthy of Heaven.
Argument from Theistic Coherence.
Claim: God is all-good because he is omnipresent (everywhere), and all-powerful (nothing is inaccessible to him), and all-knowing (omniscient), so nothing can hide from him, including the truth. And when a person knows all the facts about what is right to do, and nothing can prevent them acting, they must be all-good. Therefore the idea of God is logically necessary and coherent.
Response: If God is omnipotent he can create a wall. If God is omnipotent he can create a wall that is too hard to knock down. If God is omnipotent he can create a wall that even he can't knock down. If he can't create such a wall he is not omnipotent. If he can't knock it down, he's not omnipotent. Therefore omnipotence is logically impossible. If God is omniscient he knows what he will do in the future. If God knows what he will do in the future, he can't change his mind. If he changes his mind, he didn't see the future clearly and therefore is not omniscient. If he can't change his mind he is not omnipotent. Therefore God is not logically coherent.
Prima Causa argument.
Claim: Everything has a cause. And those prior causes have prior causes. Therefore, there is an infinite chain of causes going back in time. At some point, if we are to explain the universe (have a complete explanation), we have to stop at one particular cause and say this was the First Cause. That is God. (St Aquinas).
Response: The first cause could have been a big bang, or a quantum event. It did not have to be God. Indeed a big bang or quantum event is simpler to explain than an infinite being with plans, thoughts, moral values, etc. Furthermore, quantum random events do not have causes, so most atomic events are uncaused causes. Yet they are not God. If you say "God did it", you are still left with "Why?". The precise initial cause - be it a Big Bang, a Cosmic Bounce, etc., are all up for further research.
Claim: Perfect things necessarily contain the property of existence. Anything that is perfect necessarily exists. God is perfect. Therefore God necessarily exists. (St Anselm).
Response: I can imagine a perfect unicorn. Perfect things necessarily exist. Therefore at least one unicorn exists. We can do this with any item. I can imagine a perfect dragon, or a perfect flying car. We therefore must reject the claim that perfection entails existence, otherwise we can wish anything into existence.