Here's a brief and very accessible explanation of one of the most powerful arguments for the existence of a being which, if it does exist, would be very much like the God of traditional theism.
The argument is called the Kalam version of the cosmological argument, and, as the name suggests, it has roots in both Islamic and Christian thought. The argument has experienced something of a rebirth since the acceptance in the latter half of the twentieth century of the standard Big Bang model of the origin of the universe and has been especially prominent in the work of philosopher William Lane Craig.
Notice a couple of things. The argument is not a proof that God exists. It simply points out that both premises are more reasonable to believe than their alternatives and, since the conclusion follows from the premises, it's more reasonable to believe it than it is to believe its alternative - i.e. that the universe has no cause.
Of course one can argue that even if the universe has a cause that cause is not necessarily the God of traditional monotheism. This contention becomes less plausible, however, when we inquire as to what we can reasonably infer about this cause from the effect (the universe) it has produced.
As the video points out, the cause would have to transcend space and time since these are physical constituents of the universe. It would also have to be very powerful to be able to cause a universe to come into being, and, we might add, it would have to be very intelligent to design a universe that exhibits the amazing mathematical precision ours does. We might also think that, since the universe contains rational, personal beings (us), it's reasonable to suppose that the ultimate cause of such beings is itself rational and personal.
If these inferences are valid then we're very close to the existence of a being not unlike the God of Christianity, Judaism, and Islam.
Even so, one attribute traditionally imputed to God that we cannot derive from this chain of reasoning, as far as I can see, is his goodness. The universe is morally ambiguous and so to arrive at the conclusion that its creator is omnibenevolent requires other resources and arguments.
Nevertheless, establishing the reasonableness of believing that there exists a being that possesses the attributes that a creator of this universe must have is not insignificant. In fact, in my opinion, it brings us so close to the God of monotheism that it makes the denial of the existence of such a being, seem intellectually perverse.