Russian President Vladimir Putin has said he will do everything possible to bring to justice those who committed the "vile and cynical" murder of opposition politician Boris Nemtsov.
In a telegram to Mr Nemtsov's mother, published on the Kremlin website, Mr Putin offered condolences and praised Mr Nemtsov's openness and honesty.
Mr Nemtsov was shot four times in the back on a bridge near the Kremlin.
Western leaders demanded a transparent investigation into the killing.
In the telegram to Mr Nemtsov's 86-year-old mother, Dina Eydman, Mr Putin said: "We will do everything to ensure that the perpetrators of this vile and cynical crime and those who stand behind them are properly punished."
He said: "Please accept my deepest condolences in connection with this irreparable loss. I sincerely share your sorrow.
"Boris Nemtsov has left his mark in the history of Russia, in its political and public life. He occupied significant posts in a difficult time of transition in this country. He always openly and honestly voiced and upheld his views."
Expressing shock at the "cruel and cynical murder", Prime Minister Dmitry Medvedev said Mr Nemtsov was a "principled person" who "acted openly, consistently and never betrayed his views".
On Saturday there was a steady stream of people leaving flowers at the site of the killing.
Mr Nemtsov, 55, served as first deputy PM under President Boris Yeltsin in the 1990s but fell out of favour with Mr Putin and became an outspoken opponent, particularly on the Ukraine conflict.
In a recent interview, Mr Nemtsov had said he feared Mr Putin would have him killed because of his opposition to the war.
Mr Nemtsov died hours after appealing for support for a march on Sunday in Moscow against the conflict.
The march, due to be held in a Moscow suburb, has now been cancelled, and the organisers have been given permission to hold a mourning procession in the centre of the city.
Russian state media say it will begin on Kitaigorodsky Proezd at 15:00 local time (12:00 GMT) and pass the site of the killing. Analysts say it is rare for state media to announce the time and place of opposition rallies.
Amid widespread global outrage, US President Barack Obama condemned the killing as a "brutal murder".
Analysis: Artyom Liss, BBC Russian
Russia woke up in shock on Saturday. The press, the social media, the politicians - all describe the killing of Boris Nemtsov, one of the leaders of the country's opposition, as something that was - until Friday night - completely unthinkable.
He was gunned down a stone's throw away from the Kremlin, in an area which is always tightly policed, and where security cameras are everywhere you look. He was, it appears, tracked for hours as he travelled around central Moscow.
The investigation will probably take a very long time. From experience, few in Moscow believe that it will name those who ordered the killing. The Russian police have a long history of catching people who pull triggers - but they are much less successful when it comes to identifying their masterminds.
The Russian government must conduct a "prompt, impartial and transparent investigation", the US president urged.
"I admired Nemtsov's courageous dedication to the struggle against corruption in Russia and appreciated his willingness to share his candid views with me when we met in Moscow in 2009," Mr Obama said in a statement.
A statement from the office of German Chancellor Angela Merkel spoke of Mr Nemtsov's "courage" for his frequent criticism of Russian government policy.
Mrs Merkel "calls on President Putin to ensure that the murder is cleared up and the perpetrators brought to justice", her spokesman Steffen Seibert said.
UK Prime Minister David Cameron echoed the calls for an inquiry, saying he was "shocked and sickened" by the news.
Ukrainian President Petro Poroshenko described Mr Nemtsov as a friend of Ukraine.
He said: "Boris had declared he would provide clear evidence of Russian armed forces' participation in [the war] in Ukraine. Somebody was afraid of this... They killed him."
Amnesty International demanded a "prompt, impartial and effective" investigation into what it said was "a cold-blooded murder of one of those free voices whom the authorities have so actively sought to silence".
Dmitry Peskov, a spokesman for Mr Putin, told the BBC that any suggestion the Kremlin was involved in the killing was nonsense.
Mr Nemtsov was shot at around 23:40 (20:40 GMT) on Friday while crossing Bolshoy Moskvoretsky Bridge accompanied by a woman, Russia's interior ministry said.
He was shot with a pistol from a white car which fled the scene, police said.
Russian investigative committee head Vladimir Markin said in a statement that several motives for the killing were being considered including "Islamic extremism" and the victim's alleged links with Ukraine.
"Mr Nemtsov may have been sacrificed by those who do not shun anything to reach their political gains," the statement said.
It added that the attack was meticulously planned and the killers had been tracking Mr Nemtsov's movements around the city.
Violent deaths of Putin opponents
April 2003 - Liberal politician Sergey Yushenkov assassinated near his Moscow home
July 2003 - Investigative journalist Yuri Shchekochikhin died after 16-day mysterious illness
July 2004 - Forbes magazine Russian editor Paul Klebnikov shot from moving car on Moscow street, died later in hospital
October 2006 - Investigative journalist Anna Politkovskaya shot dead outside her Moscow apartment
November 2006 - Former Russian spy Alexander Litvinenko died nearly three weeks after drinking tea laced with polonium in London hotel
March 2013 -Boris Berezovsky, former Kremlin power broker turned Putin critic, found dead in his UK home