In this paper, we consider the automatic performance tuning of dense vector and matrix-vector operations on GPUs. Such operations form the backbone of level 1 and level 2 routines in the Basic Linear Algebra Subroutines (BLAS) library and are therefore of great importance in many scientific applications. As examples, we develop single-precision CUDA kernels for the Euclidian norm (SNRM2) and the matrix-vector multiplication (SGEMV). The target hardware is the most recent Nvidia Tesla 20-series (Fermi architecture). We show that auto-tuning can be successfully applied to achieve high performance for dense vector and matrix-vector operations by appropriately utilizing the fine-grained parallelism of the GPU. Our tuned kernels display between 25-100% better performance than the current CUBLAS 3.2 library.