Nicola Sturgeon has pledged to tackle the so-called "cybernats" over online abuse, saying SNP members who "cross the line" will face disciplinary action.
The Scottish First Minister made clear it is "not acceptable" for people to use social media to "threaten violence, or hurl vile abuse, or seek to silence the voice of others through intimidation".
The SNP leader said her party would take action against members if postings on social media breached the party's code of conduct or online guidance .
She called on other leaders to follow her stance.
Writing in the Scottish Daily Mail, Ms Sturgeon said: "When tweets or postings from SNP members that cross the line are brought to our attention, we will act - as we have done before.
"That is why I am making clear today that the SNP will take steps to warn those whose behaviour falls short of the standards we expect.
"We will tell them to raise their standards of debate, to stick to issues not personalities, and to ensure robust and passionate debate takes precedence over abuse and intemperate language.
"I am also making clear that where appropriate we will take disciplinary action. In the SNP we have a code of conduct and online guidance for our members.
"Where that code is broken, members should have no doubt that we will use our disciplinary processes."
Ms Sturgeon has faced calls at Holyrood to crack down on the perpetrators of online abuse in her party.
A row erupted in the run-up to the general election after Neil Hay, the SNP candidate in Edinburgh South, described unionists as ''quislings'' - a word which originates from the name of a wartime Nazi collaborator - on his ''Paco McSheepie'' Twitter account.
Mr Hay, who also mocked elderly voters, was one of the few SNP candidates not elected in May, with Labour's Ian Murray holding the seat.
In March an SNP member was suspended from the party after directing foul-mouthed homophobic abuse at Scottish Conservative leader Ruth Davidson on Twitter.
The abuse, which was littered with expletives attacking Ms Davidson's sexuality, was posted on the Twitter profile Laird O'Callaghan, under the username @SparkyBhoyHH.
A number of famous Scots, including Harry Potter author JK Rowling and cycling star Sir Chris Hoy have been subjected to online abuse in recent years.
Rowling, a Labour supporter who donated money to the campaign to keep Scotland in the UK, has been branded a ''traitor to Scotland'' and ''Blairite scum''.
In the run-up to the independence referendum Sir Chris was called ''a traitor'' and a ''typical Scots Tory naysayer'' after he spoke out about the lack of training facilities in Scotland.
Tennis star Andy Murray was subjected to online abuse last year, after he declared his support for the Yes campaign in the independence referendum.
On the day of the vote he tweeted it was a "huge day for Scotland" and said: "Let's do this!''
A Twitter user responded: "'Wish u had been killed at Dunblane, you miserable anti-British hypocritical little git. Your life will be a misery from now on.''
Ms Sturgeon, who has 229,000 followers on Twitter, said the level of abuse she receives on social media "would make people's hair stand on end were they to see it".
She added: "I choose simply to ignore it, but that doesn't mean that online comments which cross the line of decency are acceptable.
"Where political disagreement is passionate, open, honest and robust and conducted with respect, it is welcome. Even where views are expressed using language that I wouldn't use, I accept that - after all that's in the nature of free speech."
She urged other political leaders to follow her example, saying: "We must send out a clear message that politics in Scotland will not be sullied by this behaviour.
"Raising the standard of debate is a responsibility across the board and I urge all parties to do as we have done - to say clearly that crossing the line will not be tolerated."