Let $g(n,;l_1,;l_2,;d,;t,;q)$ be the number of general4-regular graphs on n labelled vertices with $l_1+2l_2$ loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with five types of properties to determine the asymptotic behavior of $g(n,;l_1,;l_2,;d,;t,;q)$ and hence that of g(2n), the total number of general 4-regular graphs where $l_1,;l_2,;d,;t;and;q;=;o(sqrt{n})$, respectively. We show that almost all general 4-regular graphs are 2-connected. Moreover, we determine the asymptotic numbers of general 4-regular graphs with given connectivities.