More than half of the Scottish population is not on the register, despite research showing 90% of people are in favour of donation.
Public Health Minister Michael Matheson will today meet 29-year-old Samantha Bell, who underwent a life- saving heart transplant last year.
The campaign is encouraging the public to make their wishes known to loved ones and to sign up as a donor.
Currently, 40% of families refuse to allow permission for organs to be donated after death.
The Evening Times is campaigning for an opt-out system of organ donation to be introduced in Scotland, following the lead of Wales.
This would means a default position where everyone is a donor though the right to refuse – opt out – is respected.
Campaigners believe a change in the law is the best way to drive up numbers along with other changes to the infrastructure.
Research shows countries with an opt-out in place tend to have higher donation rates and a lower family refusal rate.
Our drive is backed the the British Medical Association and major charities including the British Heart Foundation, Kidney Research UK and the Cystic Fibrosis Trust.
More than 10,000 people have signed our petition.
In a major boost for our campaign, a debate will be held later this week at the Scottish Parliament calling for a change to opt-out.
A motion has been put forward by Paisley-born SNP MSP Kenneth Gibson recommending the move.
The motion has so far been backed by 38 MSPs including Glasgow MSPs Bill Kidd (SNP) Patricia Ferguson (Labour) and Patrick Harvie (Greens).
Mr Gibson said: "It is time for presumed consent and I will making that point strongly in the debate, given the pressing need for more donors - the Evening Times has mounted a strong campaign."
More than 43 people died in Scotland last year waiting for a transplant.
The campaign launched today will see NHS Greater Glasgow and Clyde encouraging families to discuss their organ donation wishes.
A Scottish Government spokeswoman said: "This year's campaign has been designed to get people talking about organ donation to make their wishes known to their loved ones and sign up to the NHS Organ Donor Register."
HOW TO GET BEHIND OUR CAMPAIGN
WE need your help to make this happen and save more lives. Sign our petition to lend your support to an 'opt-out' system for organ donation at www.evening times.co.uk
There is also a printable version of the petition available on our website.
We need as many of you as possible to distribute it in community centres, churches, mosques, leisure centres and public places so we can achieve our target of 10,000 signatures.
You can also sign up on Twitter and Facebook via http://bit.ly/etoptin. And don't forget to send the link to all of your friends and relatives.
In the meantime, to join the existing national organ donor register, text 'ETDONATE' to 61611.
Texts are charged at the usual standard network rate (any texts sent in reply are free of charge to the recipient). You can also join the list by calling the NHS Donor Line on 0300 123 23 23. Lines are open 24 hours a day. Calls are charged at your contracted rate for local calls.
Those who wish to register online can also do so by filling out a form on www.organdonationscotland.org/nhs_organ_donor_register.html