It is the story of many American cities. Despite decades of efforts, school districts have made little progress on desegregation.

In dozens of places, racial divides have grown more pronounced. A growing wave of litigation — including the affirmative action cases involving Harvard and the University of North Carolina last summer — has challenged the fairness of policies intended to create more diverse student bodies.

But recently, New York City has become the unlikely focus of renewed efforts to reduce racial segregation in public schools.

This spring, a lawsuit challenging selective admissions in New York’s schools, where huge numbers of Black and Latino students attend schools that are less than 10 percent white, was allowed to move forward in the state courts. It recalls an earlier set of court cases that tried to uproot separate and unequal academic tracks for Black and Latino students — and that argued that segregated schools and classrooms denied their rights.

The tactics for attacking segregation today are different from those employed in school districts generations ago. Across the city, some local districts and schools are trying other ways to mix student bodies without explicitly using race, as a way to avoid limits created by past Supreme Court cases.

Here are four things to know:

Since Brown v. Board of Education prohibited racial segregation in public schools seven decades ago, a robust body of research has shown that racially and economically diverse public schools can benefit all children.