In the last three decades, memory safety issues in system programming languages such as C or C++ have been one of the most significant sources of security vulnerabilities. However, there exist only a few attempts with limited success to cope with the complexity of C++ program verification. We describe and evaluate a novel verification approach based on bounded model checking (BMC) and satisfiability modulo theories (SMT) to verify C++ programs. Our verification approach analyses bounded C++ programs by encoding into SMT various sophisticated features that the C++ programming language offers, such as templates, inheritance, polymorphism, exception handling, and the Standard Template Libraries. We formalize these features within our formal verification framework using a decidable fragment of first-order logic and then show how state-of-the-art SMT solvers can efficiently handle that. We implemented our verification approach on top of ESBMC. We compare ESBMC to LLBMC and DIVINE, which are state-of-the-art verifiers to check C++ programs directly from the LLVM bitcode. Experimental results show that ESBMC can handle a wide range of C++ programs, presenting a higher number of correct verification results. Additionally, ESBMC has been applied to a commercial C++ application in the telecommunication domain and successfully detected arithmetic-overflow errors, which could potentially lead to security vulnerabilities.