Let \(\Omega\) be a smooth bounded domain in m-dimensional Euclidean space \({\mathbb{R}}^ m\) with boundary \(\Gamma =\Gamma_ 0\cup \Gamma_ 1\), with \(\Gamma_ 0\) possibly empty and \(\Gamma_ 1\) nonempty and relatively open in \(\Gamma\). The authors consider the initial-boundary value problem \[ (1)\quad y_{tt}(x,t)=\Delta y(x,t)\quad (x\in \Omega,\quad 0\leq t\leq T) \] \[ (2)\quad y(x,0)=y^ 0(x),y_ t(x,0)=y^ 1(x)\quad (x\in \Omega) \] \[ (3)\quad y(x,t)=0\quad (x\in \Gamma_ 0,\quad 0\leq t\leq T) \] \[ (4)\quad D_{\nu}y(x,t)=v(x,t)\quad (x\in \Gamma_ 1,\quad 0\leq t\leq T) \] (D\({}_{\nu}\) is the outer normal derivative on \(\Gamma)\). The exact controllability problem for (1)-(4) is whether there exists some \(T>0\) (depending on \(\Omega\), \(\Gamma_ 0\), \(\Gamma_ 1)\) such that for all initial data \(y^ 0\), \(y^ 1\) in a given product space \(Z=Z_ 1\times Z_ 2\) of functions defined in \(\Omega\) there exists a control function v in a given space V of functions defined in \(\Gamma\) \(\times (0,T)\) such that the corresponding solution of (1)-(4) satisfies \(y(x,T)=y_ t(x,T)=0\) (x\(\in \Omega).\) The main results in this paper are on the following cases: (a) \(Z=H^ 1_ 0(\Omega)\times L^ 2(\Omega)\), \(V=L^ 2(\Gamma_ 1\times (0,T))\), and (b) \(Z=L^ 2(\Omega)\times (H^ 1_ 0(\Omega))'\); \(V=(H^ 1(0,T;L^ 2(\Gamma_ 1)))'\), where, besides smoothness assumptions, the triplet \(\{\Omega,\Gamma_ 0,\Gamma_ 1\}\) is assumed to satisfy geometrical conditions expressed in terms of a general vector field; here \(H^ 1_ 0(\Omega)\) is the subspace of \(H^ 1(\Omega)\) consisting of all functions that vanish on \(\Gamma_ 0\). In the Neumann case \(\Gamma_ 0=\emptyset\) there are controllability results with \(Z=L^ 2(\Omega)\times (H^ 1_ 0(\Omega))'\) without geometric conditions but for a special class V of controls larger than \(L^ 2(\Gamma \times (0,T))\).