In 2011 there were 16,230 youngsters in care - a total which rose each year since 2001, when the number was 10,900.
While the overall number of children in care has jumped 49% in that time, the number of those children cared for by grandparents or other relatives in formal kinship care arrangements is almost four times higher than 10 years ago.
According to the study for the organisation Parenting across Scotland there were 980 children in formal kinship care in 2001. This rose to 3910 by 2011.
The report says: "In terms of children in need of care, the most striking change between 2001 and 2011 was the marked increase in the number of looked-after children."
It added that while most children in care stay in the family home, there was a trend of children being looked after in other community settings, such as staying with friends or relatives.
This was seen as being "indicative of increasing levels of kinship care".
But the report warned kinship carers could themselves be a vulnerable group, as most are grandparents who suffer high levels of disability.
In 2011, 9% of Scots - some 500,000 people - were providing unpaid care for a friend or family member struggling as a result of disability, illness or age.
While the proportion of people who are carers is the same as in 2001, increasing numbers of carers are having to devote more of their time to this role.
More than two-fifths of carers provide 20 hours of care a week or more, up from 37% in 2001, while 27% spend 50 hours a week or more looking after someone, compared to 24% a decade ago.
The proportion of men who are classed as being "economically inactive" because they are looking after the home or family increased from 1% to 3% over the decade, but nearly six times as many women perform this role, with 17% of females "economically inactive" for the same reason in 2011.
The report, which draws on census data from 2001 and 2011, found the number of households in Scotland increased by 200,000 to 2.4 million over this period.
An increase in the number of people living alone meant single-person households were the most common living arrangement in 2011, accounting for more than a third of all households.
The trend towards smaller households also saw a rise in families with one dependent child, with the report finding almost a third of children in 2011 were growing up in a home without siblings.
Parenting across Scotland - an umbrella body made up of charities including Children 1st, Children in Scotland and One Parent Families Scotland - will launch the report at a conference on Tuesday where Children's Minister Aileen Campbell will speak.
Clare Simpson, project manager at Parenting across Scotland, said: "Families are juggling multiple responsibilities and providing large amounts of care while continuing to be active in the economy.
"Women are almost six times as likely as men to be looking after the home and family, and are also much more likely to be working part-time.
"To ensure families are able to flourish in these hard pressed times, it is imperative we ensure the right supports are in place to allow parents to work. High-quality flexible childcare which parents can afford, and family-friendly working practices, are crucial."
Marion Davis of One Parent Families Scotland said: "The report on families in Scotland will make a valuable contribution to ensuring parents and children receive the support and services they need."