Their online subscriptions to music or film should still be available when they travel and they should be able to buy online content not easily accessible from a home provider: Europe's creative output is one of its richest resources, and those who want to enjoy it should be able to pay to do so, even when it is only sold in another Member State.
At the same time, Europe needs to maintain choice and diversity by protecting intellectual property in a way that ensures a flourishing and innovative creative sector. Our enforcement of the intellectual property regime must have teeth.
When businesses raise their prices unfairly on the basis of the nationality of a consumer, their Internet Protocol or physical address, consumers suffer - and trust in the market decreases.
Prices of course can differ legitimately, for example because of sales tax or delivery costs. However, there should be no extra charges for making a payment simply because an online retailer is in one Member State and your card issuer in another, and national rules on promotions should not be allowed to stop consumers from one country taking advantage of an online offer that others can access.
There ought to be transparency about pricing and businesses shouldn't unfairly exploit consumers' trust in the single market.
It can be daunting enough to buy cross-border without worrying that your parcel won't be delivered quickly or that a rogue business will mean it's impossible to get help if there's a problem.
The EU needs a common set of consumer rights tailored to the purchase of digital content.
These - and Europe's other consumer protections - need to be easy for consumers to understand and act on, and properly enforced by all Member States working together.
The EU must also find creative ways of making it easier for businesses to meet their obligations, as it can be difficult to understand the legal requirements of a consumer's home market.
Data is the lifeblood of the digital economy. We all benefit from greater insight into consumer behaviour, which drives innovation. But consumers need a data protection framework they can trust if they are to be willing for their data to be used to unlock new and better services.
We need to be prepared for the next evolution in the digital economy, which will be driven by consumers making choices about how to use their own data. People must be able to move their data from one service provider to another, or use it themselves.
Of course new technologies bring new risks, so we need to set data protection in a broader framework that ensures the security of citizens. But the proliferation of data is inevitable. If we don't create the right climate for seizing the opportunities this brings, we can be confident that data-driven innovation will continue elsewhere and simply be sold into the EU.
In a mobile world, being charged a fortune to receive emails or use apps when there's supposed to be a single market feels unfair. SMEs also need to be free to do business on the move.
But the end of roaming charges must not create a disproportionately difficult environment for the challenger operators that are keeping home prices low. The freedom for consumers to use their devices anywhere must go hand in hand with the freedom for operators to operate anywhere.
This means we need infrastructure access for operators at a reasonable price via a reduction in wholesale costs.
Putting 25 public services online at https://www.gov.uk is forecast to save £250m a year.
By digitising services and requiring consumers to submit data less frequently, public services could be provided more efficiently and more securely for everyone.
Consumers should be able to use their digital identity to prove that they are who they say they are in a safe and secure way, confident that their data is protected when accessing online services in all Member States.
Where barriers prevent this also being available in the private sector, we should build on the existing regulation on digital identity to tackle these.
These tools have the potential to give consumers access to banking, transport and other services online without having to go through paper-based processes.