The justices are expected to start work Monday by denying many of the nearly 2,000 appeals that piled up in recent months. The court also is hearing argument in a bankruptcy dispute and an appeal by criminal defendants seeking shorter prison terms.
During the new term, the court will look at provocative anti-gay protests at military funerals and a California law banning the sale of violent video games to children. These cases worry free speech advocates, who fear the court could limit First Amendment freedoms.
The funeral protest lawsuit, over signs praising American war deaths, "is one of those cases that tests our commitment to the First Amendment," said Steven Shapiro, legal director of the American Civil Liberties Union.
Another case involves a different aspect of the First Amendment, the government's relationship to religion. The justices will decide whether Arizona's income tax credit scholarship program, in essence, directs state money to religious schools in violation of the constitutional separation of church and state.
Under Chief Justice John Roberts, marking his fifth anniversary on the court, and with the replacement of Justice Sandra Day O'Connor by Justice Samuel Alito, the court has been more sympathetic to arguments that blur the line between government and religion, as long as one religion is not favored over another.
Kagan, confirmed in August, is the one new face on the court, but nearly everyone will be sitting in different seats when the term opens.
Like so much else at the Supreme Court, the justices sit according to seniority, other than the chief justice at the center of the bench. The retirement of John Paul Stevens, who had served longer than the others, means Roberts now will be flanked by Justices Antonin Scalia and Anthony Kennedy.
Kagan and Justice Sonia Sotomayor, who joined the court last year, will sit at opposite ends of the bench. The woman with the longest tenure, Justice Ruth Bader Ginsburg, also is now the senior liberal-leaning justice with Stevens gone.
Though it's never certain how changes will affect the court's direction, President Barack Obama said he was looking for someone in the mold of the liberal-leaning Stevens when he chose Kagan. If Kagan votes as Stevens did, her presence would not affect the ideological divide that has four justices on the conservative side, four on the liberal side and Kennedy in the middle, though more often with the conservatives.
Then, too, a justice's first term is not necessarily an accurate predictor of future performance. If anything, getting a read on Kagan in her first year may be even harder because her former job as Obama's solicitor general already has forced her to take herself out of 24 of the 51 cases the court has so far agreed to hear. The solicitor general is the top lawyer who argues the government's cases before the high court.
The first case from which she is withdrawing will be argued Monday, and Kagan will slip out of the courtroom before Roberts invites the lawyers to begin their argument.
Kagan's absences create the potential for the eight remaining justices to split 4-4 in some cases. That outcome leaves in place the decision reached by the most recent court to have the case, but leaves unsettled the issue the high court was set to resolve.
A second Arizona law, imposing penalties on businesses that hire illegal immigrants, also is before the court this term. At issue is whether the state law intrudes into an area, immigration, that really is the federal government's responsibility.
The result at the Supreme Court could signal how the court might resolve another suit working its way through the federal courts over the Arizona immigration law that puts local police officers on the front lines of enforcing federal immigration law, said Brian Wolfman, a Georgetown University law professor.
Several cases that pit consumers against business also revolve around when federal law trumps state action.