At 1220 GMT, Russia's dollar-denominated RTS index was up 1.27 percent to 1,235.38 points, having reached 1,238.26 earlier in the session, its highest since September 2014.
The rouble-based MOEX Russian index, previously known as MICEX, was 1.14 percent higher at 2,232.54 points, slightly down from a session high of 2,242.87, its highest since January last year.
"Inflation expectations globally are being priced in aggressively, which in turn has brought big price moves in commodity-exporting countries and currencies, placing Russia on investor radars," Alfa Bank analyst John Walsh said in a note.
The pace of production expansion and growth in Russia is the best among emerging countries, he said, adding that Russian stocks remained cheap while offering attractive dividends.
The rouble gained 0.87 percent to trade at 68.08 versus the euro, its highest level since November 2017, reflecting the situation on the global markets.
The European currency dropped on Tuesday after slipping from last week's high as investors were cautious after a months-long rally.
The rouble was also near 4-month highs against the dollar, gaining 0.1 percent to trade at 57.06.
Brent crude oil, a global benchmark for Russia's main export, was up 0.31 percent at $67.99 a barrel. Brent hit $68.29 a barrel earlier in the session, its highest since May 2015.
Many Russians returned to work on Tuesday after Russia's Christmas and New Year holidays, but some traders were still away from their desks.
"The Russian rouble is quite positively positioned before the return to full trading on the local market. The rouble may break through its September limit versus the dollar, falling below 56.80," analysts at Sberbank CIB said in a note.