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This may sound extremely obvious, but it's nonetheless worthwhile to make sure you're already running as fast as you can. This is especially pertinent for PC users, as the Caps Lock button can mean the difference between walking and running. Keeping the Caps Lock button activated should keep you running while you're using the directional keys. For console systems, ensuring you're pushing the control stick as far as it can go will maximize your speed. Again, this may sound painfully obvious, but there are rare cases where a controller may not be working to one's complete satisfaction. While it doesn't necessarily affect your real speed, if you're trying to run somewhere fast, you should pay attention to the smartest way to get there. Running up steep surfaces won't get you somewhere nearly as fast as smooth pavement. Outside of Washington DC, Fallout 3's open world is generally pretty easy to navigate, so try to make the most direct route to wherever you're going. At the same time, you should take care to avoid enemy-heavy areas; if you're trying to move as efficiently as possible, getting into fights will only serve to slow you down. As is true for all Bethesda RPGs, the type of armour you wear will have a notable effect on your movement speed. A light armour, like leather armour, will keep you moving briskly. Better still, you can go without armour completely for maximum speed.  Also keep in mind that armour tends to have a big weight associated with it. Not only will wearing a heavy armour slow you down, the weight attached will bog down your weight capacity considerably. On the other hand, some sets of armour like Power Armour come with a Strength bonus, which in turn will help boost your weight capacity. One of the most common speed-related complaints in Fallout 3 is the speed penalty you get from carrying too much equipment. If you're finding your speed bogged down, try dropping or selling some of your supplies. Before too long, you should find your character running at a better rate. Aim to discard junk items and extra sets of armour, as they tend to be the heaviest items in the game. Your character will be slowed down naturally if you have any weapons out. Try to make a habit of holstering your weapons whenever you're not using them. A hefty portion of the game's not going to require constant use of weapons, so while you might want that extra bit of caution, in most cases it's unnecessary.  You can holster your weapon by holding the Reload button. Keep in mind that two-handed weapons (e.g. rifles, flamethrowers) will slow you down more than one-handed weapons, like pistols. If you're still worried about downtime while you're drawing a weapon, you can get the Quick Draw perk, which will make the holster/draw process 50% faster.
Make sure you're running at your full speed. Plan your paths accordingly. Dress your character in lighter armour. Keep your weight capacity as low as possible. Keep your weapons holstered.