In the UK children all get 250 pounds invested for them at birth (if you are below a certain income level you get an extra 250) in a child trust fund.
The money becomes available to the child at eighteen until then the parents can choose from a variety of different funds, from completely safe savings funds to completely shares based funds. The government recommends a form called a "stakeholder account" which is a shares based account with certain regulations about spreading of assets, levels of charges and also, the money is gradually moved from shares to cash starting when the child reaches thirteen.
Parents etc are allowed to add up to 1200 pounds per year more to these accounts as they wish. They are tax free.
We have chosen not to add more to our child trust funds, but instead they each have a "bare trust" for which me and my mother are trustees (neither of us are tax payers, so it makes things easier). and we put extra money in those. Bare trusts have very few protections in law - if my mother and I agreed to embezzle the money we could, but we'd have to agree to it, and that's not likely! They're invested in stocks and shares (through F and C) at the moment, but we intend to use the stakeholder fund model of gradually moving it to safer investments from when they reach 13 yrs old. When they are eighteen the money will be put into their names, but I hope by then to have taught them to be responsible with it and use it to get them through university and possibly as down payment on a house when they graduate.
We put in their share of the child benefit every month and my mother has split some of her savings between her grandchildren and that went in as a lump sum. Any other extra money goes in as and when they get it - my brother gave them some money at birth.
This is almost identical to what was done for me by my parents and grandparents, and I really appreciated it.