I went to a talk last night with the local APM (Association of Project Managers) branch - well worth it!
Sue Vowler, who was one of the pilots and early adopters for PRINCE2, wrote the book on P3O (published 2008), and explained the thinking behind it.
P3O is based on the very best of principles - how can we take organisations which struggle to keep up with the ever-changing world and help them to deliver faster, more consistently, for lower cost?
Sue gave the example of Vodafone, one of the companies which kicked it off. 23% of new projects succeeded, and the time to bring a new product to market was between 9 and 18 months. They decided to standardise processes, to use best practice not just when the right person was in place as project manager, but every time, and to ensure that business cases were reviewed in the context of both the business and the other initiatives underway, rather than each in isolation on its own merits. They changed the business completely - 80% of projects succeeded, and new product to market time came down to 3 months!

Essentially, Sue said, companies do the right thing, and/or do things right. They might do a lot of things right, but they might be doing the wrong things - result is failure. Lots of effort into auditing and processes and templates, high quality, but simply going in the wrong direction or going in too many inconsistent directions.
They might be doing the right things, but only sometimes, and sometimes they know what they should do but don't do it.
To succeed, organisations and people need to both do the right thing, and do it right.
Which means three functions for the P3O:
Firstly: this process can be applied to any type of organisation, the approach just has to be appropriate. So a small organisation might just have a couple of people who co-ordinate all of the project managers in the organisation, without taking them away from where they are.
A larger organisation may decide to locate all of the project and programme managers in the same office, preferably near to the Chief Exec (who they should report to), and best of all to put all of their dashboards (red / amber / green and actions needed to get to success) on the walls so the Chief Exec and any staff coming into meetings can look and say "oh yes I know how to sort that one out"
A Multi-national or large government department can implement a hub-and-spoke, with a central office coordinating, and local offices by function or by region.
The first question to ask is "how much do you spend on projects?". Most organisations know down to the last penny how much they spend on Business as Usual (BAU), and how to save money. But they often don't have a clue how much they spend on projects, and that is often why most of it is going to waste.
"How many project managers do you have? How many portfolio managers? Do you use a single project / programme management methodology or lots of different ones? Do you have too many templates and do people not use them? Who do the project managers report to? (if IT Director then you are hardly out of the starting gate)"
Sue reminded us that most organisations score 1 out of 5 (the worst) for maturity on Benefits Management, and it agrees with my own assessment of most of the organisations I've encountered - but then if they didn't need help they wouldn't ask me in. Having said that, to recognise you need help puts you between 2 and 3 on the maturity model for Benefits Realisation/ Business Benefits Management.
I hope that this quick overview is useful and look forward to comments - Hugo