Abstract: We present coordinated multiwavelength observations of the bright, nearby BL Lacertae object Markarian 421 (Mrk 421) taken in 2013 January-March, involving GASP-WEBT, Swift, NuSTAR, Fermi-LAT, MAGIC, VERITAS, and other collaborations and instruments, providing data from radio to very high energy„ (VHE) gamma-ray bands. NuSTAR yielded previously unattainable sensitivity in the 3-79 kiloelectronvolt range, revealing that the spectrum softens when the source is dimmer until the X-ray spectral sh…