Some have opted for various versions of a priori ontological arguments: arguments in which it is claimed that God's existence follows necessarily from the very meaning of the concept of God. Just like it is impossible for a bachelor to be married (as he would then no longer be a bachelor), it is claimed that it's impossible for God not to exist, since the very concept of God implies his/her existence.
Others have opted for a posteriori arguments based on empirical evidence. In the cosmological argument, the general idea is that, given its contingency, the very existence of the universe, regardless of its particular properties, requires some sort of explanation. In the teleological argument it is claimed that the particular properties of the universe we inhabit (its complexity and apparent fine-tuning for life) cannot simply be attributed to chance but to a conscious and personal designer who has intelligence and forethought.
In his posthumous masterpiece Dialogues Concerning Natural Religion, the always fascinating philosopher David Hume set out to show, with great eloquence and wit, that these proofs, especially the argument from design, are not nearly as convincing or as strong as they may appear at first glance. Unlike Plato or Bishop Berkeley, who are well known for giving all the best lines of their dialogues to their mouthpieces (and pretty crappy ones to their detractors), Hume spread his powerful insights concerning the nature and limits of knowledge, causal inference and religion over all three main characters, though it is clear that Philo represents most closely Hume's own cautious skepticism. If you've never had a chance to read this classic work, sit back and enjoy the following beautiful dramatization:
Introduction: Pamphilus to Hermippus: