Convex records have an appealing purely geometric definition. In a sequence of d -dimensional data points, the $n$th point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the bivariate (i.e. two-dimensional) setting. For iid (independent and identically distributed) points, we establish an identity relating the mean number ⟨ $R_n$ ⟩ of convex records up to time $n$ to the mean number ⟨ $N_n$ ⟩ of vertices in the convex hull of the first $n$ points. By combining this identity with extensive numerical simulations, we provide a comprehensive overview of the statistics of convex records for various examples of iid data points in the plane: uniform points in the square and in the disk, Gaussian points and points with an isotropic power-law distribution. In all these cases, the mean values and variances of $N_n$ and $R_n$ grow proportionally to each other, resulting in the finite limit Fano factors $F_N$ and $F_R$ . We also consider planar random walks, i.e. sequences of points with iid increments. For both the Pearson walk in the continuum and the Pólya walk on a lattice, we characterise the growth of the mean number ⟨ $R_n$ ⟩ of convex records and demonstrate that the ratio $R_n$ / ⟨ $R_n$ ⟩ keeps fluctuating with a universal limit distribution.