The first, serendipitous, radio-astronomical observations by K. Jansky were at decametric wavelengths. However, after the initial pioneering work, long-wavelength radio astronomy was largely abandoned in the quest for higher angular resolution because ionospheric structure was thought to limit interferometric imaging to short (< 5 km) baselines. The long-wavelength (LW, 2 - 20 m or 15 - 150 MHz) portion of the electromagnetic spectrum thus remains poorly explored. The NRL-NRAO 74 MHz observing system on the Very Large Array has demonstrated that self-calibration techniques can remove ionospheric distortions over arbitrarily long baselines. We describe the scientific justification and initial technical design of the Low Frequency Array (LOFAR) -- a fully electronic, broad-band antenna array operating in the 15 - 150 MHz range with a collecting area of 1 km2 at 15 MHz. The longest baselines may be 500 km, providing an angular resolution of 10' at 15 MHz and 1' at 150 MHz. The combination of large collecting area and high angular resolution will enable LOFAR to produce images with sensitivities of order 1 mJy at 15 MHz and 300 (mu) Jy at 150 MHz. As such LOFAR will represent an improvement of 2 - 3 orders of magnitude in resolution and sensitivity over the state of the art. A key operational goal of LOFAR will be solar observations -- both passive imaging and radar imaging. In the latter mode LOFAR will serve as the receiver for bi-static observations of the Sun, with particular emphasis on the imaging of coronal mass ejections. LOFAR will serve as an astrophysical laboratory to study the origin, spectrum, and distribution of the Galactic cosmic ray electron gas and as an instrument to probe the high-redshift Universe.