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The Zeeman effect and dust grain alignment are two major methods for probing magnetic fields (B-fields) in molecular clouds, largely motivated by the study of star formation, as the B-field may regulate gravitational contraction and channel turbulence velocity. This review summarizes our observations of B-fields over the past decade, along with our interpretation. Galactic B-fields anchor molecular clouds down to cloud cores with scales around 0.1 pc and densities of 104–5 H2/cc. Within the cores, turbulence can be slightly super-Alfvénic, while the bulk volumes of parental clouds are sub-Alfvénic. The consequences of these largely ordered cloud B-fields on fragmentation and star formation are observed. The above paradigm is very different from the generally accepted theory during the first decade of the century, when cloud turbulence was assumed to be highly super-Alfvénic. Thus, turbulence anisotropy and turbulence-induced ambipolar diffusion are also revisited.