This paper concerns the general problem of investigation of bitopologies in the sense of Kelly on some sets of mappings \(f:X\to Y\) of arbitrary fixed sets \(X\) and \(Y\) under those or other conditions. The authors consider two such bitopologies on the set \(C(X, Y)\) of all bicontinuous mappings \(f: (X, \tau_1, \tau_2)\to (Y, \tau_1', \tau_2')\), namely the so-called 2 compact open bitopology and the bitopology of quasi-uniform convergence on 2 compact sets. A set \(K\subset X\) will be called 2 compact if \(K\) is a compact set in the topological space \((X, \tau_1\vee \tau_2)\). The 2 compact open bitopology \((T_1, T_2)\) is determined as follows. For \(i= 1, 2\) the set of all sets \([K, G_i]= \{f\mid f(K)\subset G_i\}\), where \(K\subset X\) is a 2 compact set and \(G_i\in \tau_i'\), generates the topology \(T_i\). The bitopology of quasi-uniform convergence can be determined if \((X, \tau_1, \tau_2)\), \((Y, \tau_1', \tau_2')\) and a quasi-uniform structure \({\mathfrak U}\) on \((Y, \tau_1', \tau_2')\) are given. In this situation one can determine the quasi-uniform structure on \(C(X, Y)\) by a base consisting of sets \(\{(f, g)\mid (f(x), g(x))\in U\) for all \(x\in K\}\), where \(K\) is a 2 compact set and \(U\in {\mathfrak U}\). Such a structure is a quasi-uniform structure on \(C(X, Y)\) so it determines the corresponding bitopological structure on \(C(X, Y)\), which is called bitopological structure of 2 compact quasi-uniform convergence. The authors present some results connected with the notions of 2 compact open bitopology and bitopology of 2 compact quasi-uniform convergence. In particular they present a characterization of 2 compact quasi-uniformizable spaces in terms of such bitopologies and a necessary and sufficient condition for (bicomplete) quasi-pseudometrizability of 2 compact open bitopology.