The unfinished business includes a Texas sodomy law and the question of whether the government can force public libraries to filter the Internet. And after that: the possibility the court will see its first retirement in nearly a decade.
June usually is the court's busiest month, or at least the time when its doings make the most headlines. This year the justices are due to hand down rulings in more than 20 cases, including the two that will govern how universities may consider an applicant's race in deciding admissions.
Although the Supreme Court already has narrowed the use of affirmative action in other areas, supporters say banning it for top colleges would mean a quick return to almost all-white classrooms.
The court heard from hundreds of interest groups, professors, students, business leaders and even retired military leaders supporting the programs in place at the University of Michigan and its law school.
The court's statement is expected to guide other consideration of race in public life, and opponents of affirmative action hope it will all but end the practice in the public sector.
Absent such a potentially momentous decision, the year's marquee case would have been Lawrence v. Texas, a challenge to a law criminalizing gay sex. The court may be prepared to reverse an unpopular 16-year-old ruling in a Georgia case that upheld similar laws.
"For the gay community, Lawrence is their Brown v. Board of Education, their major civil rights case," said Georgetown University law professor Richard Lazarus.
Most states have repealed anti-sodomy laws. Where they still exist, they are rarely enforced. Yet the laws undermine gay equality in other areas and are fundamentally unfair, the lawyer for two gay men told the court when the case was argued in March.
The case began when police burst into a Houston bedroom and arrested the two for a sex act that would have been legal for a heterosexual couple.
The public library case returns the court to the clash between free speech on the Internet and the need to protect young children from smut and sexual predators.
Congress has passed three laws governing children's safe use of the Internet since 1996, but the Supreme Court struck down the first and blocked the second from taking effect.
The latest measure, signed by President Clinton in 2000, requires public libraries that receive federal technology funds to install filters on their computers or risk losing aid.
A three-judge federal panel ruled the Children's Internet Protection Act violates the First Amendment because the filtering programs also block sites on politics, health, science and other nonpornographic topics.
Sneaker maker Nike Inc. also has a free speech case before the justices. The company claims it is being unfairly muzzled in its attempt to fend off allegations of exploitative labor practices in the Third World.
Big business is watching that case closely, as are media and public relations companies. If the high court should rule against Nike, it could become easier for consumer activists or others to truth-squad company statements and sue over alleged falsehoods.
The insurance industry is awaiting the court's ruling on a California law intended to help people recover money from unpaid Holocaust-era insurance claims. The state wants insurers to disclose information about old policies, but insurers say that is a daunting, expensive and legally risky task.
Other subjects before the court include campaign donations by certain interest groups; a prelude to a major campaign finance ruling expected next fall; and new restrictions on family visits to state prisoners. Michigan prison authorities say the restrictions protect children, while lawyers for prisoners say the policy is unconstitutional.
The court also will decide whether a mentally ill dentist can be medicated against his will so he can stand trial for fraud and whether states can erase statutes of limitations to revive prosecutions in old child molestation cases.
Retirement rumors are swirling around the court as the session winds down. None of the nine justices has announced plans to leave, but Chief Justice William H. Rehnquist and Justice Sandra Day O'Connor are considered good bets to retire soon. If either should choose to leave this year, it would give President Bush his first opportunity to choose a justice.