  The main goal of this expository article is to survey recent progress on the
arithmetic Siegel-Weil formula and its applications. We begin with the
classical sum of two squares problem and put it in the context of the
Siegel-Weil formula. We then motivate the geometric and arithmetic Siegel-Weil
formula using the classical example of the product of modular curves. After
explaining the recent result on the arithmetic Siegel-Weil formula for Shimura
varieties of arbitrary dimension, we discuss some aspects of the proof and its
application to the arithmetic inner product formula and the Beilinson-Bloch
conjecture. Rather than intended to be a complete survey of this vast field,
this article focuses more on examples and background to provide easier access
to several recent works by the author with W. Zhang [LZ22a, LZ22b] and Y. Liu
[LL21, LL22].
