Big question, but a good one. Here are some tips to get you started:
Firearms and Bows -- firearms and bows need to travel as checked baggage (not carried on the plane) in a locking, hardside, airline approved case.
The case must be locked in such a way to prevent even partial opening that would allow a hand to slip inside (i.e. if the case has multiple closures each must be locked). Firearms and magazines must be unloaded. Ammunition should be separate from the firearm. Those are the universal rules, but individual airlines and countries have their own additional regulations in some cases. For example, some require that the firearms also wear trigger locks in transit. Some require the gun be as disassembledd as possible for travel (i.e. bolt removed, shotguns broken down, etc.). Some have limits on the number of guns that can be transported in a single locked case. Do some web surfing prior to packing to check the regulations of the airlines you'll be using and the countries in and through which you'll be traveling.
Ammunition -- besides the fact that you can't bring ammo on board as carry-on -- it too must be checked -- the biggest thing to remember is that no loose ammunition in allowed on airlines even in checked baggage. That means no rounds left in the pocket of a hunting coat or even in a magazine. Airlines and TSA prefer that ammo is in it's original hardside container, but any kind of hardside container that separates individual cartridges is acceptable. Ammo in such a container can go in your regular checked luggage or pack as long as it's locked, too. There are overall weight limits on the amount of ammo that can be checked. In North America, that's 2 kilos or about 11 pounds.
Knives -- though changes are underway regarding what can be carried on commercial airlines, rest assured you won't be allowed to carry any kind of knife useful for hunting on board. Check your knives. They can ride in your gun or bow case, just make sure they are securely sheathed or wrapped so they don't do any damage to the other contents of the case.
Optics -- because they are high value but relatively small and concealable, optics are prime targets of luggage theives. Other than the scope securely mounted to your rifle, plan to carry on your optics including binoculars, spotting scope, rangefinder and cameras. In addition to being able to safeguard this expensive equipment, you also save it from the abuse that's sometimes doled out by baggage handlers. You may end up explaining and showing what all the equipment is at the x-ray, but it's well worth the extra hassle to make sure your optics arrive in good shape ... and arrive at all!
Survival Kits, Fire Starters, Gun Cleaning Supplies -- Airline security is picky about some small things when it comes to chemicals. If they discover solvent or gun oil in your gear, expect that these will be confiscated from your bags. You definitely cannot carry them on. Components in your survival and first aid kits will get scrutiny as well -- if it can be used to easily start a fire in the woods, then there's the suspition it could be used to start a fire on board as well.
Muzzleloading and Black Powder -- it's flat out against the law to fly with black powder, bulk propellants, primers and or percussion caps. They are explosives. If you try to fly with them and are caught, you can expect the same treatment terrorist bombers receive! Do not do it. If you are going to be muzzleloader hunting at a distant location, make plans to acquire the components you need at that location. Bullets and sabots are okay to have in your checked baggage, but you'll have to do some explaining to uninformed airline and security agents. To many of them the terms "bullet" and "cartridge" are the same.
Weight Limits -- As airlines increase baggage fees, they also become much pickier about baggage wieght limits. Prior to packing, contact the airlines you'll be flying to find out what they allow per bag. Some are 40 pounds, some 50 and some 75. Some will even allow 100 pounds, but for a significantly higher price. Cost per bag and weight limits are also somewhat determined by the class in which you're flying and an your status with that airlines as a frequent flyer.
For travelling by airlines in North America, the best on-line source to begin your research and answer additional questions is: www.tsa.gov.