Brussels is a small city of around 1 million people, and while bad things can happen anywhere, there are no real no-go areas. However, locals consider the main stations, especially the Gare du Midi, to be a bit of a danger zone - a particular problem for tourists is being distracted from looking after your belongings, while consulting a map or guidebook or trying to figure out a ticket machine. Don't put a wallet in your back pocket or on the top of a bag, pickpockets operate in particular on the metro and at the Midi Market, where they can take advantage of crowds.

The following neighborhoods, as can be seen on the map, are best avoided for tourists especially during the night : Anneessens, Anderlecht,  Chaarbeek, Brussels North, St-Josse, Marollen and Molenbeek. 

Tourists face the usual pickpocket-ting problems at airports, train stations and crowded streets around the central square and the Mannekenpis. 

After dark, surface transport may feel safer than certain metro stations, so you might want to get to grips with buses and trams.  Make sure you know what the registered taxis look like.  If you are going to walk somewhere e.g., hotel to station it's worth familiarising yourself with the walking route and perhaps have a look at it on street view.  E.g. Gare du Nord in Brussels doesn't look like a station from the outside!  This helps in looking purposeful and confident which also puts off thieves.

Rue des Bouchers is a narrow pedestrian street off the main square. It's a tourist trap; locals don't eat there. There are lots of over-priced fish restaurants, and the staff from the restaurants stand on the street in your way trying to get you to eat at their restaurant. There are lots of distractions and lots of tourists - and therefore lots of pickpockets. One of the few decent restaurants on the street "Au Vieux Bruxelles" has a warning sign on the door as you leave - "Watch out for pickpockets!" - the street is notorious for them.

Brussels is currently under a Level 3 high threat level which means that there is a possible and probable threat against a person,  group or event according to Belgian authorities. The possibility of a terrorist attack is real and travelers are advised to consult both the US Embassy's site and the US State Department's travel advisories.