Asian equities mostly closed higher after Wall Street finished a volatile third quarter Monday on a positive note.
The pound rose against the dollar, boosted by UK Prime Minister Boris Johnson saying his government would "shortly" present the European Union with new Brexit deal proposals.
Markets were meanwhile looking ahead to the end of the week.
"The US jobs report on Friday is naturally front of mind," said Craig Erlam, senior market analyst at Oanda trading group.
Investors are also looking ahead to next week's resumption of trade war talks between Beijing and Washington, said Quincy Krosby, chief market strategist of Prudential Financial.
The latest discussions set for October 10 are "clearly a positive for the markets because the effect it has on the world economy is paramount going into the last quarter", Krosby said.
Before then, China on Tuesday began a week-long break to celebrate the 70th birthday of the People's Republic.
Sydney's stock market closed up 0.8 percent and the Australian dollar dipped after the Reserve Bank of Australia announced it had cut interest rates to a historic new low.
Amid fears about the flagging domestic economy, the central bank slashed rates by another 25 basis points to 0.75 percent.
Resource-rich Australia dodged the worst of the financial crisis but the economy recently recorded its weakest annual growth in a decade, expanding just 1.4 percent in the year to June.
It is also feeling the impact of US-China trade tensions, which have rattled the global economy as central banks worldwide attempt to offset economic risks by lowering rates.
The World Trade Organization on Tuesday cut its 2019 trade growth forecast to 1.2 percent, a sharp downgrade on the 2.6-percent rise predicted in April, which the body blamed on escalating trade tensions.
"The darkening outlook for trade is discouraging but not unexpected," WTO director general Roberto Azevedo said in a statement.
Elsewhere, "oil prices are recovering slightly after falling almost four percent on Monday on reports that Saudi Arabia has fully restored output at the facilities that were attacked only two weeks ago", said Erlam.
"There was a lot of scepticism around whether they would manage to fully restore these facilities but this remarkable turnaround has apparently been delivered."
- Key figures around 1100 GMT -
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London - FTSE 100: DOWN 0.5 percent at 7,375.14 points
Frankfurt - DAX 30: DOWN 0.2 percent at 12,404.36
Paris - CAC 40: DOWN 0.2 percent at 5,664.50
EURO STOXX 50: DOWN 0.2 percent at 3,562.27
Tokyo - Nikkei 225: UP 0.6 percent at 21,885.24 (close)
Hong Kong (closed) - Hang Seng at 26,092.27
Shanghai (closed) - Composite at 2,905.19
New York - Dow: UP 0.4 percent at 26,916.83 (close)
Euro/dollar: UP at $1.0902 from $1.0900 at 2050 GMT
Pound/dollar: UP at $1.2308 from $1.2294
Dollar/yen: UP at 108.26 yen from 108.10 yen
Brent North Sea crude: UP 1.2 percent at $59.95 per barrel
West Texas Intermediate: UP 1.1 percent at $54.66 per barrel