Some people forgo the joy of two wheels and take the cage instead because they have to take things with them—groceries from the supermarket, a laptop to work, a life-size statue of a tyrannosaurus rex... We can't help you with the T-rex, but for most other items there's a motorcycle-sized solution just waiting for you. Here's how you can have it all.
This is the simplest solution. You probably already have one. Just load it up, strap it on your back, and ride off into the sunset. Case closed.
Okay, it's not really an ideal solution. The biggest disadvantage is that all the weight rests on the rider, not the bike. This can lead to fatigue and achy muscles. It also prevents you from moving around while you ride, whether you're hanging off the bike in the corners or simply stretching. Optimally, the bike should carry the weight, not you. You could strap the backpack to the seat or a sissy bar if you have one (we'll get into more about straps later). Backpacks are still rather small compared to some other possibilities out there.
If your gas tank is metal, a tank bag is probably the next easiest solution. Most stick magnetically, so just slap it on and go. Even without metal, many include straps that you can use to secure the bag to your bike. This gets the weight off your body and carries about the same amount of cargo as a backpack. Some tank bags even turn into backpacks to make them easy to carry off the bike as well. One big advantage of a tank bag is that you can access your stuff while you're riding. Don't try that with a backpack.
Just about any bike has room for a pair of saddlebags tossed on the back. Some may require a luggage rack, while others just sit on top of your passenger seat. You can carry much more in saddlebags than a backpack or tank bag. You could easily pack for a weekend trip with a set of saddlebags like these. Just make sure they fit your bike correctly and don't sag into contact with the chain or back wheel.
Instead of, or in addition to, any of these options, you can add a trunk to the back of your bike. Givi and other manufacturers make them, but on my KLR 650 I simply bolted a knock-off Pelican case to my rear rack and called it good. It's watertight and good enough for commuting. Mine isn't easily removable, but the Givi trunk I had on my Honda PC800 popped off with a push of a button. It even had a handle so that I could carry it like a suitcase, which is how I often used it during my longer travels.
Speaking of my PC800, one of its best attributes was its integrated cargo area. I could carry a 24-pack of beer on either side of the back wheel, as well as other very important items. Most bikes don't include their own cargo areas, but some do, providing an easy solution.
Either included with the bike, like the Honda Gold Wing, or added from the aftermarket, this is a good way to go for weatherproof, secure storage. It locks, unlike a soft bag, but is often removable unlike my PC800's integrated storage. This is the best solution for serious cargo hauling.
There is one exception, however. While many adventure bikes come with big boxy side cases, those can pin or crush your legs if you drop the bike, which can be a common occurrence off-road (ask me how I know). Some riders prefer soft luggage for this since it squishes when you fall.
If you don't have luggage, or are carrying an item too large or awkwardly shaped to fit in luggage (your next set of tires, a guitar, a recurve bow, etc.) you can always strap it to the back. Bungee cords are the traditional favorite, but better solutions exist, like Rok Straps. These attach more securely, can be tightened regardless of the strap length (unlike a bungee cord), and even separate in the middle, allowing you to retrieve your precious cargo without digging under everything to find the hook. Speaking of which, make sure you hook any strap you use to secure mounting points, like luggage racks or the frame, not flimsy bits of plastic that will break.
The Sky's The Limit
That's not enough space for you? You can attach a sidecar. You can tow a trailer. You can even do both. Why do we even have cars anymore? Motorcycles can do everything a car can do, and more.