We will begin by introducing the use of higher-order Fourier analysis in additive combinatorics for a general audience. In particular, we will discuss the arithmetic regularity lemma and how it identifies a certain class of arithmetically-structured functions -- nilsequences -- as extremal objects for problems in additive combinatorics. We will then discuss how it has recently come to light that the analysis of nilsequences on certain arithmetic patterns -- those which satisfy a certain algebraic criterion known as the flag condition -- is easier than the general case, and discuss some recent ideas and developments in overcoming the difficulties that arise when the arithmetic pattern of interest is not flag.