We present comprehensive studies on thin diffraction lenses made of arrays of subwavelength, parallel nanoslits in a gold film. Such a nanoslit lens can operate either as a conventional convex or concave lens. The lenses can be designed to focus linearly polarized light with polarization either perpendicular (TM-lens) or parallel to the slits (TE-lens), while the orthogonal polarization diverges when passing through the lens. The designs of each lens are initially built on the dispersion relations for wave propagation through a parallel-plate waveguide. Both TM-and TE-lenses were realized experimentally, and full-wave numerical simulations fully support the experimental results.