The authors prove the following results. Theorem A: A complete noncompact Riemannian manifold \(M\) of positive minimal radial curvature has exactly one end. Theorem B: If the minimal radial curvature of \(M\) is bounded below by 1 and if the volume of \(M\) is greater than 3/4 \(\omega_ n\), where \(\omega_ n\) is the volume of the unit \(n\)-sphere \(S^ n\), then \(M\) is homeomorphic to \(S^ n\). Theorem C: If \(M\) has non-negative minimal radial curvature with base point at \(0\in M\) and satisfies \(\lim_{r\to\infty} {\text{vol }B(0,r)\over b_ 0(r)}> {1\over 2}\), where \(b_ 0(r)\) is the volume of the \(r\)-ball of a complete and simply-connected Riemannian \(n\)-manifold of constant zero curvature, then \(M\) is diffeomorphic to \(\mathbb{R}^ n\). Theorem D: For given constants \(\kappa\in \mathbb{R}\) and \(n\geq 2\) there exists a constant \(\varepsilon_ 0= \varepsilon_ 0(n,\kappa)> 0\) such that if \(M\) has the properties \(\dim M= n\), \(k_ M\geq -\kappa^ 2\), \(k_ 0\min\geq 1\) and \(\text{vol}(M)\geq \omega_ n- \varepsilon\) for \(\varepsilon\in (0,\varepsilon_ 0)\), where \(k_ M\) is the sectional curvature of \(M\), then there exists a diffeomorphism \hbox{\(f: M\to S^ n(1)\)} with the property that there exists a constant \(\tau= \tau(n,\kappa,\varepsilon)> 0\) with \(\lim_{\varepsilon\to 0}\tau= 0\) and \(e^{-\tau}\leq{\| df(\xi)\|\over \|\xi\|}\leq e^ \tau\) for every non-zero tangent vector \(\xi\) to \(M\).