Three men were shot dead after killing seven and injuring 48 more with a van and knife attack in the UK capital on Saturday night.
Many parties have suspended national campaigning on Sunday, however Prime Minister Theresa May said that the General Election on Thursday will go ahead.
First Minister Nicola Sturgeon chaired a meeting the Scottish Government’s Resilience Committee on Sunday where she was updated on the situation following the London attack.
She heard that there is currently no indication of Scottish victims or intelligence of any specific threat to Scotland, yet Police Scotland will be active at relevant transport hubs and will work in partnership with the British Transport Police to ensure any potential witnesses who may be returning to Scotland are identified.
Police Scotland is also increasing its visibility including increasing the number of Armed Response Vehicles.
She said: “My heartfelt sympathies - and those of everyone in Scotland - are with the families of those who lost their lives during last night’s despicable and cowardly terror attack.
“Our thoughts are also with those who sustained injuries. We wish all of them a full and speedy recovery.
“I also want to thank our emergency services. Last night we saw yet again the bravery, dedication, selflessness and professionalism of the police and of those who work in the NHS and fire service.
“This morning I was briefed by the deputy National Security Adviser and updated both on what is currently known about the attackers and also on the reasons for the Joint Terrorism Analysis Centre (JTAC) decision to keep the security threat level, at this stage, at severe.
“I also chaired a meeting of the Scottish Government’s Resilience Committee and received a full briefing from Police Scotland.
“At this point in time, we have no information to suggest that any of the victims of this attack are from Scotland - however, it is not yet possible to be definitive about that and we are liaising closely with relevant authorities in London as more information becomes available.
“Our police and health services have also offered any practical assistance to London that may be required.
“As was the case after the Manchester attack, the public can expect to see a more visible police presence, particularly in busy areas – this will include an increase in armed police.
“At what I know is an anxious time, my message to the public is to be vigilant, report anything of suspicion to the police - but otherwise continue to live your lives as normal.
“I have no doubt that - just as we saw after previous attacks, most recently the attack in Westminster - the people of London will pull together, in all of their diversity, and refuse to be cowed by acts of terrorism.”
The UK threat level remains at severe following the attack, and Police Scotland said it “continues to dynamically review all safety and security plans and operations”.
Assistant Chief Constable Steve Johnson said: “Police Scotland stands with our colleagues and the communities of London following the terrorist attack on Saturday 3 June 2017.
“This comes soon after the attack in Manchester, and other recent attacks abroad. Our thoughts are with everyone who has been affected by these events. We will provide any assistance necessary to the Metropolitan Police Service.
“While we understand that the public will clearly be appalled by these events, I would urge each and every member of our communities to remain united against extremism and hate.”
He added: “Police Scotland continues to dynamically review all safety and security plans and operations. An element of this includes ensuring our armed policing and specialist resources are appropriately deployed.
“The UK threat level remains at severe. We have no specific information that Scotland is at risk of attack and I would ask you to go about your business as normal, remaining alert but not alarmed.”
He said that Police Scotland will not tolerate any attempts to target any community by “any misguided individual or group” and said that the force continues to work with partners at home and abroad to counter the threat from terrorism and ensure the safety and well-being of all communities.
He urged people to be vigilant and contact police if they suspect something is wrong.
Anyone with concerns or information about suspicious activity is asked to contact the Police on 101, the UK Anti-Terrorism Hotline on 0800 789 321, Crimestoppers on 0800 555 111, or in an emergency dial 999.
The SNP, Scottish Conservatives and Scottish Labour have suspended national campaigning on Sunday while the Scottish Liberal Democrats have also cancelled campaign visits.
Flags outside the Scottish Parliament have been lowered to half mast in the wake of the attack.
The London atrocities come less than two weeks after the terrorist bombing at the Ariana Grande concert in Manchester on Monday May 22.
Eilidh MacLeod, 14, from Barra was among 22 people who died in the attack which also left dozens injured.
Her body is being flown home to the island on Sunday ahead of her funeral on Monday.