Co-existing wireless systems, which share a common spectrum, need to mitigate out-of-band (OOB) radiation to avoid excessive interference. For legacy systems, OOB radiation is well understood and is commonly handled by digital precompensation techniques. When using large arrays, however, new phenomena and hardware limitations have to be considered. First, signals can be radiated directionally, which might focus the OOB radiation. Second, low-complexity hardware is used for cost reasons, which increases the relative amount of OOB radiation. Given that massive MIMO and millimeter wave communication rely on base stations with a large number of antennas, the spatial behavior of OOB radiation from large arrays will have significant implications for the hardware requirements of future base stations. We show that, if the OOB radiation is beamformed, its array gain is never larger than that of the in-band signal. In many cases, the OOB radiation is close to isotropic even when the in-band signal is highly directive. With the same total radiated power, the OOB radiation from large arrays is therefore never more severe than from a legacy system with the same adjacent-channel-leakage ratio. Further, the OOB radiation is less detrimental than from a legacy system since the high array gain of the in-band signal allows large arrays to radiate less total power than legacy systems. We also show how OOB radiation from large arrays varies with location in static propagation environments and how these effects vanish when averaged over the small-scale fading. Since a higher relative amount of OOB radiation can be tolerated for large arrays, the linearity requirement can be relaxed as compared to legacy systems. Specifically, less stringent linearity requirements on each transmitter makes it possible to build large arrays from low-complexity hardware.