The problems with the Cosmological Arguments
The essence of these arguments is the supposed impossibility of a series having no first term, that is, the rejection of the idea of an infinite regression. Some have argued that there is nothing illogical about an infinite regress of causes, and have stated that the cosmological argument contradicts itself because if we grant that nothing causes itself then there cannot be a first cause by definition.
An argument against the concept of an infinite regression is that if it were true an infinite amount of time must have been traversed in order to reach the present, which cannot ever be covered.
But this sounds rather like Zeno's paradoxes. [ Zeno of Elea (c 490-430 BCE)]
Mackie in Miracle of Theism (1982) supports the idea of a Prime Mover or First Cause.
He writes that even if you could conceive of a moving railway train consisting of an infinite number of railway carriages, one would still have to posit an engine at the front of it, or it simply wouldn't move.
David Hume (1711-1776) in his Dialogues Concerning Natural Religion, argued that if we accept that every effect has a cause, then the causal chain is infinite.
Firstly, the concept of a First Cause of an infinite series is logically contradictory, because if it is first, then it is in time, and therefore the
series cannot be infinite. In tracing an eternal succession of objects it seems absurd to inquire for a general cause or first author.
Secondly, the argument that seeks to find a single cause of all caused things, a First Cause of the causal chain, involves a fallacy. As Hume writes, if he were to show you the particular causes of each individual in a collection of twenty particles of matter, 'I should think it very unreasonable should you afterwards ask me what was the cause of the whole twenty.'
This is a fallacy that Bertrand Russell was later to emphasise, and which is now known as the Quantifier-shift Fallacy.
Bertrand Russell says Aquinas moves in his Second Way from saying "Everything is changed by something" to "Some one thing changes everything". Which is like saying that because everyone in the human race has a mother the human race itself must have a single mother.

Immanuel Kant (1724-1804) himself argued that causation was a term that could only meaningfully be used about the material world, and that one could not transfer it beyond that into the metaphysical. The cosmological arguments all start with observable reality and move out of it into areas that we do not know about. In what sense are our existing logical concepts able to deal with, let alone find out about, what "happened" before the beginning of the universe?

The principle of the conservation of energy/matter, which is generally accepted among physicists today, would imply that the universe is infinite.
However, such physical laws are derived from observation, and are only applicable in "normal science" situations. If the universe did have a beginning, the laws of physics would only start applying after that beginning.
Some supporters of the Cosmological arguments look to the so-called "Big Bang" theory of the origin of the universe which until very recently was the virtually unchallenged orthodoxy of the science of cosmology. This theory posits a clear dateable beginning of the universe (c. 15 billion years ago), which can be discovered by extrapolating backwards the speed at which the universe is expanding.
It has been argued that God must have caused the Big Bang to start everything off.
Thereafter the laws of Physics take over - theoretical physicists have claimed that the history of the Universe is fairly clear after the first hundred thousandth of a second).
However, the First Cause could be the Big Bang itself and not necessarily God.
Also the Big Bang Theory has been challenged. Stephen Hawking in his book A Brief History of Time suggests that time might be curved like a globe and that therefore the universe did not have a beginning in a clear "singularity" at all.

Other problems with the Cosmological arguments
1.How do we know that everything has a cause? Our notion of causality may simply be a psychological one, imposed on the data we perceive.
Post hoc does not necessarily mean propter hoc. ('After' does not mean 'on account of'.)
2.How do we know that items cannot be self-causing?
3.Even if the causal chain is finite, how do we know that it starts with a single cause? It may start with a multiplicity of causes. After all, most events in life have multiple causes.
4.It may be the universe itself rather than God that is non- contingent.
5.Aquinas' Third Way goes from saying that all material things can not- exist to saying that they therefore must at some time not have existed. This is not necessarily a logical step. As Brian Davies says, "My cat is kickable, but does that mean I have to kick it?"
6.In his Third Way Aquinas moves from saying that because everything must have not existed, therefore there was a time when nothing existed.
It is not necessarily the case that because all things are contingent that there must have been a time when nothing existed; there could be an infinite regress of overlapping finite existences.
7.The Third Way involves the concept of a "necessary being". Hume argues that such a being could not be susceptible to proof, as whatever can be imagined as existing can be imagined as not existing , and therefore we cannot posit necessary existence of anything.
8.Aquinas rather assumes that the Prime Mover, the First Cause, etc are what people generally understand by God; but this is a very questionable assumption. There is nothing in the concept of The First Cause, even assuming that it is an intelligible concept, that requires it to be personal, intelligent, forgiving, loving, etc.
9.The Fourth Way is an argument that clearly owes much to Platonic ideas of perfection and participation (ie things are good if they participate in a perfect goodness). The problem with this argument is to show that there actually are objective degrees of perfection, without resorting to the concept of an absolute self-existing perfect Being (which is assuming what you are being asked to prove)
10.Clearly Leibniz' argument is subject to the criticism in 8 above; but also the suggestion that everything must have a sufficient reason is a very questionable assumption. Indeed for most atheists this is the exactly what they deny.
There is a kind of special pleading though: when we ask questions about why the universe exists some theists say the answer is because of God.
The question must then arise: why does God God exist?
As Bertrand Russell put it: 'the universe is just there, and that's all there is to say.'
Others say that if Bertrand Russell and others refuse even to sit down at the chess- board and make a move, they cannot, of course, be checkmated.