It will be the first time Gorsuch dons a black robe as a justice, and his presence brings the court back to full strength more than a year after the sudden death of Justice Antonin Scalia.
It is a busy week as the justices will decide whether to add some important cases for next term -- they will announce opinions Tuesday and Wednesday and they will hear arguments for the last scheduled sitting of the term.
All eyes will be trained on the court's most recent addition, the 49-year-old Gorsuch, to see how he interacts with his new colleagues and to try to glean clues about his jurisprudence.
Gorsuch joins the court as it rounds the final turn of a term set to conclude by the end of June.
Last week, the justices held their regular closed-door conference to discuss pending petitions. After the conference concluded, the court announced that Gorsuch had not participated presumably because he needed time to adjust to his new job and prepare for upcoming oral arguments.
The results of the conference will be announced Monday morning at 9:30 a.m. ET.
One pending petition concerns a Colorado cake artist who refused to make a cake for a same-sex couple's wedding reception, citing religious objections.
Another case deals with North Carolina's voter ID law
that was blocked from going into effect before the last election by the lower courts. The justices could also decide whether to hear a case concerning claims by undocumented Central American women and their children who were apprehended immediately after entering the United States surreptitiously in late 2015. Lawyers for the families seek to challenge their expedited removal proceedings in federal court arguing they face gender-based violence at home.
Also on Monday, the court might see some emergency petitions stemming from a controversy concerning efforts in Arkansas to block several scheduled executions in the coming days. Lawyers for death row inmates argue that the proposed method of execution constitutes cruel and unusual punishment.
Starting at 10:00 a.m. ET on Monday court watchers will hear arguments in three cases, including Town of Chester v. Laroe Estates. While the case is a technical one dealing with the scope of the standing doctrine, it is being argued by former Acting Solicitor General Neal Katyal, who served during the Obama administration. Katyal introduced Gorsuch during his confirmation hearing and his endorsement infuriated some of his fellow Democrats who fought against Gorsuch.
It is unlikely Gorsuch would recuse himself from the case, as justices hear arguments all the time from lawyers with much closer ties than Katyal and Gorsuch.
On Wednesday, the court will hear arguments in Trinity Lutheran Church v. Comer. The religious liberty case has the potential of becoming the term's most important case.
It was granted shortly before Scalia's death, but the justices held off on setting it for argument, perhaps cognizant that it might need a tie-breaking vote.
It involves a daycare facility run by the Trinity Lutheran Church in Missouri. The church sought a state grant given to facilities that use recycled tires as a surface for playgrounds to improve safety.
Missouri awarded the grants to other nonprofits, but said that the daycare facility was ineligible because the Missouri Constitution bars funding to churches. Lawyers for the church say the state's action violates the Constitution. If the court rules in favor of the church, it could narrow the separation between church and state.
Last week, however, the justices asked parties to address a new development in the case.
Missouri Gov. Eric Greitens recently announced his administration would reverse course and allow religious organizations to apply for and be eligible for such state grants.
In a statement he said that before he came into office in 2016, "government bureaucrats were under orders to deny grants to people of faith who wanted to do things like make community playgrounds for kids."
"That's just wrong," he said.
Although Greitens said he didn't expect his decision to impact the Supreme Court case, the justices are asking the parties to respond to whether the case is now moot.
On Tuesday and Wednesday, the court will issue opinions.
There are several cases that have yet to be decided, which could indicate the justices are evenly split and may vote to have the case re-argued before the full nine-member court.
One case that seemed close during oral argument concerns a 15-year-old Mexican national who was shot to death in 2010 by a US Customs and Border Patrol agent standing on American soil
. The teenager was cowering behind a pillar just across the border in Mexico.
The family of Sergio Hernandez is seeking to sue the border official for their son's death. They say the agent violated Hernandez's constitutional rights.
The violent shooting was caught on cellphone video and sparked outrage because of the fact that Hernandez was unarmed. The case comes with the backdrop of tension between the Trump administration and Mexico over the issue of building a wall between the two countries.
Another case, Jennings v. Rodriguez, grapples with whether it is unlawful to subject thousands of immigrants fighting deportation to long-term detention without individualized bond hearings. The case is brought by a class of immigrants who seek hearings to prove that they are neither a flight risk nor a danger to society.
And Sessions v. Morales deals with an immigration statute that applies citizenship requirements for a child born of a US citizen father more stringently than a US citizen mother when they are unwed and their child is born abroad.