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The last few years have seen a flury of new results about surfaces in 4-dimensional spaces. In this talk, we will start by discussing various kinds of surfaces and some basic questions about them, like what it means for two of them to be the equivalent. We will then discuss two ways to describe smooth surfaces in 4-space explicitly, give a number of examples illustrating recent results, and mention some still-open questions. The talk will end with a sketch of some of the techniques involved. Most of the results in the talk are due to other people; a few are joint with Sucharit Sarkar.