Thalys is definitely the fastest way of getting from Brussels to Paris.
You can buy tickets directly at the Thalys website.
You can also buy them at the Belgian ( SNCB ) or French ( SNCF ) railway companies.
The prices are in principle identical.
The trip by bus takes around 5 hours, to be compared to the 1h20 of the train trip.
Apart from Eurolines you can use iDBUS .
The prices are more or less the same.
I have checked for a departure in 2 days.
Eurolines is 25 EUR and iDBUS varies between 15 and 33 EUR depending on the departure time.
Thalys starts at 79 EUR on that particular day.
In case you are unable to choose among the two bus companies, note that they are using different terminals in Brussels and Paris.
Choose whichever suits you best.
It is possible to travel by "classical" trains from Brussels to Paris.
But this is neither a real bargain nor is it really fast, as documented in another post .
It takes around 5h30, involves three train changes (Tournai, Lille and Amiens) and costs around 65 EUR (45 EUR if you are less than 26 years old).
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The fastest route is the Thalys train that takes 1:22 hours and is a non stop train.
If you compare that to the car, you need more than 3 hours according to Google .
Consequently, buses can't be any faster than cars.
Another option would be flying, but a flight takes around 1 hours, without including the time for getting to the airport, security check, waiting for departure, etc.
Other options like walking, taking a boat, use a bicycle, etc.
are not really useful options.
So to answer the first part of your question: If you want to use the fastest connection, then take the train!
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In 2010, I hitchhiked from Brussels to Paris, first from a gas station on the periphery of the city to a rest stop on the highway towards Paris, then in a fast Mercedes all the way to Paris, in fact much faster than legal.
That day I continued to Bordeaux .
Of course, hitchhiking is not reliably the fastest, but it can certainly be very fast and it is certainly the cheapest .
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If you look for cheaper and not quicker, there is also Megabus which offer prices starting for 15€.
You can compare price for that line here: www.comparabus.com/en/bus-Bruxelles-Paris-3-10
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Regarding your bus question.
Recently a new bus service emerged, called IDbus .
travel.stackexchange.com/users/None/
https://www.sobus.travel/bus-Brussels-Paris/06-12-2018#:results
You can see that the cheapest way to go from Brussels to Paris is by bus : - you can combine Ouibus and Flixbus for 10€ and you will need 5 hours- or paying 12€99 for a direct travel with Flixbus in 3 hours
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There are several ways of getting from Brussels to Paris. The fastest is probably the train, but it gets more expensive when you buy the ticket on the last minute.  My question can be really split into two:  Is there an alternative way, maybe through a different ticket operator, of buying a cheap train ticket from Brussels to Paris in short notice?  The alternative is going by Bus. What alternatives are there? Are there other operators besides Eurolines? maybe cheaper and/or faster?  Are there other options?
What is the cheapest / fastest way to get from Brussels to Paris?
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A Thalys train starts at 79 EUR on the day, and a non-stop train takes 1:22 hours. A bus will take around 5 hours, and costs 25 EUR with Eurolines or 15-33 EUR with iDBUS. Note that they use different terminals in Brussels and Paris. Megabus is cheaper but not quicker at prices starting from 15 EUR, and you can combine Ouibus and Flixbus for 10 EUR (5 hours) or take a direct Flixbus for 12.99 EUR (3 hours). Finally, hitchhiking is not as reliable, but it is the cheapest option and can be very fast. It is generally accepted that the Thalys train is the fastest Brussels to Paris connection. Buses are available and cheaper but they take longer. Flying and non-Thalys trains also take longer. Finally, hitchhiking may be faster but it depends on who picks you up.
The Thalys train is considered the fastest option for travelling from Brussels to Paris. Prices start from 79 EUR. Some bus companies provide a Brussels to Paris service. Prices are around 15 to 33 EUR but journey time is 5 hours. Other option for travelling from Brussels to Paris include hitchhiking, flying and "classical train" rather than Thalys