The Supreme Court has the final say on often highly contentious laws, disputes between states and federal Government, and appeals regarding executions.
RBG was originally appointed in 1993 by then-president Bill Clinton, and justices are appointed by a president when one either retires or dies.
Donald Trump has said he wants to nominate a successor without delay, meaning he will put in a Republican nomination – greatly upsetting the balance of the court.
The US President tweeted on Saturday he considered it an “obligation” to put forward a nominee, who if confirmed would tilt the balance of the court decisively towards the conservatives.
Mr Trump tweeted: “We [Republicans] were put in this position of power and importance to make decisions for the people who so proudly elected us, the most important of which has long been considered to be the selection of United States Supreme Court Justices. We have this obligation, without delay!”