Characterizing sound field diffuseness has many practical applications, from room acoustics analysis to speech enhancement and sound field reproduction. In this paper, we investigate how spherical microphone arrays (SMAs) can be used to characterize diffuseness. Due to their specific geometry, SMAs are particularly well suited for analyzing the spatial properties of sound fields. In particular, the signals recorded by an SMA can be analyzed in the spherical harmonic (SH) domain, which has special and desirable mathematical properties when it comes to analyzing diffuse sound fields. We present a new measure of diffuseness, the COMEDIE diffuseness estimate, which is based on the analysis of the SH signal covariance matrix. This algorithm is suited for the estimation of diffuseness arising either from the presence of multiple sources distributed around the SMA or from the presence of a diffuse noise background. As well, we introduce the concept of a diffuseness profile, which consists in measuring the diffuseness for several SH orders simultaneously. Experimental results indicate that diffuseness profiles better describe the properties of the sound field than a single diffuseness measurement.