We present here a compact 120 W, continuous-wave (CW), diode-pumped Nd:YALO laser at 1079 nm. This linearly polarized output from the optically anisotropic Nd:YALO crystal is advantageous for pumping non-linear crystals. The 1 at% doped Nd:YALO laser rod, 4 mm diameter by 97 mm long, is cut along the b-axis, with both ends anti-reflection coated at 1079 nm and 1341 nm. It is water-cooled (16 degree(s)C) and side-pumped by 5 close-coupled CW diode arrays operating at 803 nm. We obtained 121 W output at the maximum diode pump power of 571 W. The optical slope efficiency is 60%. This is, to the best of our knowledge, the highest power obtained from a diode-pumped Nd:YALO laser. We have studied this thermal lensing at different diode pump powers in lasing and non-lasing configurations. The measured thermal lens decreased from 37.5 to 11 cm as the pump power increased from 185 to 542 W in the lasing configuration. In the non-lasing case, the corresponding thermal lens decreased from 35.5 to 9.5 cm. We have also successfully operated our Nd:YALO laser at 1341 nm. We obtained a maximum output of 60 W, to the best of our knowledge, the highest diode-pumped laser output at this wavelength.