In Japan, the benchmark Nikkei 225 stock index was up by 1.4%, while Australia’s ASX 200 was 1% higher.
The Hang Seng in Hong Kong was down more than 1%. In mainland China, the Shanghai Composite Index was up by around 0.5%.
In the US on Tuesday, the Dow Jones Industrial Average, S&P 500 and Nasdaq all closed more than 1% higher.
“We could yet see some fluctuations across markets today though, particularly by assets which could be affected most by the outcome, with the US dollar and Chinese stocks being prime examples,” said Tim Waterer, chief market analyst at investment firm KCM Trade.
Donald Trump has said throughout the campaign that if he became the next US president he would dramatically increase trade tariffs, especially on China.
The former president’s more isolationist stance on foreign policy has also raised questions about his willingness to defend Taiwan against potential aggression from China.
The self-ruling island is a major producer of computer chips, which are crucial to the technology that drives the global economy.
If Kamala Harris wins investors expect her trade and foreign policies to be a continuation of Joe Biden’s more predicable approach.
Investors also have other key issues to focus on this week.
On Thursday, the US Federal Reserve is due to announce its latest decision on interest rates.
Comments from the head of the central bank, Jerome Powell, will be watched closely around the world.
On Friday, top Chinese officials are expected to unveil more details about Beijing’s plans to tackle the slowdown of the world’s second largest economy.