Electromagnetic properties of a new class of two-dimensional periodic nanostructured materials, sub-wavelength plasmonic crystals (SPCs), are investigated. An SPC is a periodic lattice of metallic inclusions with negative dielectric permittivity <0 imbedded in a dielectric host with _h>0, with the lattice period much smaller than the wavelength of light. It is found that two types of propagating electromagnetic waves are supported by SPCs: (a) scale-invariant modes whose dispersion relation is almost independent of the lattice period, and (b) scale-dependent narrow-band resonances whose dispersion strongly depends on the lattice period. The scale-invariant modes are accurately described using a frequency-dependent quasi-static dielectric permittivity _qs(ω) and a vacuum magnetic permittivity μ = 1. The scale-dependent resonances exist inside narrow frequency bands where they can have a modified magnetic permittivity . Magnetic properties originate from the non-vanishing magnetic moment produced by the currents inside any given plasmonic inclusion due to the close proximity of the adjacent inclusions. Applications of SPCs to the development of novel left-handed metamaterials in the optical range are discussed. A new paradigm of the SPC-based surface-enhanced Raman scattering is also introduced.