You will certainly¹ not need a visa. The UK has declared it will not require visas for any EU nationals for short stays, and the EU has declared the same for the UK (see European Commission press release and Brexit preparedness document). However, from 2021 you will most likely need to apply for ETIAS (an electronic authorisation similar to US ESTA). This is likely true even if there is a Brexit agreement
and possibly even if there is no Brexit (without Brexit, EU law should mean no ETIAS shall be needed).
The UK Government has issued a series of notices on the impact of a no-deal Brexit. One such notice issues specific guidance for Travelling to the EU with a UK passport if there's no Brexit deal (checked 13 December 2018). This confirms the guidance quoted in legoscias answer, but does not address a system known as ETIAS.
Once it enters into force (expected 2021), UK citizens will most likely need to apply for ETIAS, a Schengen electronic system similar to the more famous US ESTA system. See ETIAS VISA: how will it affect UK citizens:
Firstly, anytime a British citizen visiting Europe with a valid passport wishes to travel to continental Europe, prior to leaving home, they will have to apply via an online platform for an ETIAS visa-waiver.
After the UK leaves the EU, britons will be exempt from obtaining a visa, but will not be exempt from the ETIAS visa waiver.
This travel authorisation, once accepted, will be binding for 3 years for multiple visits with no limit and will become a mandatory requirement for all travelers from elegible ETIAS visa waiver countries from 2021.
UK citizens and all eligible travellers who wish to visit the Schengen Zone will need to answer a series of health and security questions when completing the ETIAS online form. It’s extremely important to answer honestly, as all the data provided in the ETIAS application will be checked against a series of databases including Europol or Interpol.
This statement from the European Commission (which the same website cites) confirms that according to stated policy, travel in both directions will remain visa free:
It would mean that UK citizens would not need a visa when travelling to the Schengen area for short stays of up to 90 days in any 180-day period. In the scenario where the UK leaves the EU without a deal, this would apply as of 30 March 2019.
This proposal is entirely conditional upon the UK also granting reciprocal and non-discriminatory visa-free travel for all EU Member States, in line with the principle of visa reciprocity. The UK government has declared its intention not to require a visa from citizens of the EU27 Member States for shorts stays for the purposes of tourism and business.
So, unless either party changes its mind and breaks with stated policy intentions, you will not need a visa. However, from 2021 you will need ETIAS.
Edit: The day after I posted this answer, The Guardian posted this "news" article on Brits needing to pay the ETIAS fee for Schengen travel after Brexit, which is old news because this was already known.
¹Of course, in theory, any government can change any policy at any time, reasonably or not, legally or not. This answer is accurate at the time of writing. Any country might suddenly close their borders after you book but before you travel. That is a risk that always exists for future travel.