Override default unhandled exception behaviour

I have this code:

#include <iostream>
#include <exception>

class TestException : public std::exception
{
public:
    char const* what() const throw() override { return msg_.c_str(); }

protected:
    std::string & message() throw() { return msg_; }

private:
    std::string msg_;
};

void ThrowIt()
{
    throw TestException();
}

int main()
{
    ThrowIt();
}

Running this when built in either Release or Debug on Windows compiled with Visual Studio results in program termination, same goes for when compiled with GCC on a Linux machine, the result is :

terminate called after throwing an instance of 'TestException'
what(): Aborted

Both terminate the program once an unhandled exception is caught. Is this behaviour strictly system specific or is this specified by the standard? Is there a cross-platform way that I can reroute every exception not handled by catch to a handler instead of just terminating the program?