The pan-European FTSEurofirst 300 index added 2.9 percent, led by banks, while Britain's blue-chip FTSE 100 index chalked up a similar gain.
MSCI's broadest index of Asia-Pacific shares outside Japan rose 1.6 percent. Japan's Nikkei climbed 2.4 percent as the yen lost ground.
U.S. e-mini stock index futures were up more than 1 percent, suggesting Wall Street would also open higher.
Two weekend polls showed "in" regaining the lead and another showed the "out" campaign's lead narrowing, though the overall picture is of an evenly split electorate. Bookmakers' odds have shown those wishing to stay in the EU ahead and Betfair put the implied probability of a vote to "remain" at 72 percent on Monday, up from 60 percent to 67 percent on Friday.
Campaigning resumed on Sunday, having been suspended for three days after British lawmaker Jo Cox was killed in the street in her constituency on Thursday.
"It might be possible that the events influenced the polls but most experts commented that it is more likely that an expected and well documented 'pull to the status-quo' is responsible for the latest swing," RBC's chief European macro strategist Peter Schaffrik said.
Sterling gained 1.64 percent to $1.4591, having hit a two-week low of $1.4013 on Thursday. It soared 2.2 percent to 152.63 yen and rose 1.1 percent against the euro to 77.72 pence.
"The outcome of the referendum is wide open again," said Ulrich Leuchtmann, currency strategist at Commerzbank. "But at least "Leave" no longer seems the most likely scenario. Should the next polls suggest that the change of sentiment persists euro/sterling may ease further."
The euro, which has also suffered due to Brexit worries, rose 0.6 percent to $1.1342, off a high for the day of $1.1382.