what is the actual law on this?

I am now confused as to what the letter of the law currently is on this: I have been under the impression, perhaps wrongly, that if you are a business and a customer buys something from you online, you are entitled to send them further emails with things like notices about events, a newsletter or special offers relating to their subject of interest. provided that all emails contain contact details and a visible 'unsubscribe' link.

I am certain this occurs all the time for me as a customer and I have come to expect it.

I now need to know EXACTLY what the laws are on this. We are based in the UK but have customers all over the world (though mostly UK and Australia)

In the US it is recommended to have people 'double opt-in'. Even when checking out you have to have them check a box that says they are willing to receive promotional emails, and then confirm their email.
I believe that the legality of it all is still up to interpretation, but according to the canspam act in the US, these are the portions that are criminal offenses:
[h=4]Criminal Offenses[/h]Although according to the law, legitimate businesses and marketers should be conscientious regarding the aspects mentioned above, there are misinterpretations and fraudulent practices that are viewed as criminal offenses:

• Sending multiple spam emails with the use of a hijacked computer
• Sending multiple emails through Internet Protocol addresses that the sender represents falsely as being his/her property
• Trying to disguise the source of the email and to deceive recipients regarding the origins of the emails, by routing them through other computers
• Sending multiple spam emails via multiple mailings with falsified information in the header
• Using various email accounts obtained by falsifying account registration information, in order to send multiple spam emails.

[h=3]Sending behavior compliance[/h]

• A message cannot be sent through an open relay
• A message cannot be sent without an unsubscribe option.
• A message cannot be sent to a harvested email address
• A message cannot contain a false header
• A message should contain at least one sentence.
• A message cannot be null.
• Unsubscribe option should be below the message.

double opt-in is definitely the way to go.