Summary: One-dimensional profiles \(f(x)\) can be characterized by a Minkowski- Bouligand dimension \(D\) and by a scale-dependent generalized roughness \(W(f, \varepsilon)\). This roughness can be defined as the dispersion around a chosen fit to \(f(x)\) in an \(\varepsilon\)-scale. It is shown that \[ D= \lim_{\varepsilon\to 0} [2-\ln W(f, \varepsilon)/\ln \varepsilon] \] holds for profiles nowhere differentiable. This establishes a close connection between the roughness and the fractal dimension and proves that \(D= 2- H\) for self-affine profiles (\(H\) is the roughness or Hurst exponent). Two numerical algorithms based on the roughness, one around the local average \(\langle f(x)\rangle_\varepsilon\) (usual roughness) and the other around the local RMS straight line (a generalized roughness), are discussed. The estimates of \(D\) for standard self-affine profiles are reliable and robust, especially for the last method.