The chromosphere is an intermediate layer of the Sun's atmosphere where radiative equilibrium breaks down. The standard chromospheric diagnostics such as the Mg II h and k and Ca II H and K spectral lines are formed under nonlocal thermodynamic equilibrium (NLTE) and they are only partially sensitive to the local conditions. Consequently, the interpretation of their profiles is not straightforward. In contrast, millimeter (mm) continuum radiation is produced by thermal free-free collisional interactions in the chromosphere under most solar conditions, and the observed brightness temperatures are better proxies for plasma temperatures. Observations at these long wavelengths have been recently enabled thanks to the Atacama Large Millimeter/submillimeter Array (ALMA), but the Sun remains largely unexplored in this spectral range.

In this thesis I explore the diagnostic potential of the mm continuum to study the solar chromosphere using inversions and radiation-magnetohydrodynamics (r-MHD) simulations. In particular, this work takes an unprecedented look at solar active-regions in the mm using some of the first solar ALMA observations.

In Paper I, we investigated whether the mm continuum helps to constrain temperatures in NLTE inversions of the MgII and CaII resonance lines using synthetic data from a 3D r-MHD simulation. In Paper II, we applied the same inversion technique to observational data in order to constrain temperature and microturbulence in plage, and we detected signatures of wave heating in coordinated observations with the IRIS satellite. In Paper III, we reported the first results of a comprehensive effort to characterize the visibility of small-scale heating events in an active-region using multiwavelength observations from the mm to the extreme-ultraviolet. We detected multiple, dynamic, transient brightenings -- we called them "millimeter bursts", and we investigated magnetic reconnection using a simulation.

This thesis shows that ALMA offers a complementary spectral diagnostic to the existing ones at visible and ultraviolet wavelengths and it underscores the importance of mm continuum observations for constraining models of the solar atmosphere.