I use shared_ptr's often.
Since Shared_ptr's are copied by-value, you can incur the cost of copying both the pointer value and a reference count, but if boost::intrusive_ptr is used, the reference count must be added to your class, and there is no additional overhead above that of using a raw pointer.
However, in my experience, more than 99% of the time, the overhead of copying boost::shared_ptr instances throughout your code is insignificant. Usually, as C. A. R. Hoare noted, premature optimization is pointless - most of the time other code will use significantly more time than the time to copy small objects. Your mileage may vary. If profiling show the copying is an issue, you can switch to intrusive pointers.
As already noted above, cycles must be broken by using a weak_ptr, or there will be a memory leak. This will happen with data structures such as some graphs, but if, for example, you are making a tree structure where the leaves never point backwards, you can just use shared_pointers for nodes of the tree without any issues.
Using shared_ptr's properly greatly simplifies code, makes it easier to read, and easier to maintain. In many cases using them is the right choice.
Of course, as already mentioned, in some cases, using scoped_ptr (or scoped_array) is the right choice. If the pointee isn't being shared, don't use shared pointers!
Finally, the most recent C++ standard provides the std::tr1::shared_ptr template, which is now on most platforms, although I don't think there is an intrusive pointer type for tr1 (or rather, there might be, but I have not heard of it myself).