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We study 3D $N=2$ supersymmetric field theories geometrically engineered fromM-theory on non-compact Calabi-Yau fourfolds (CY4). We establish a detaileddictionary between the geometry and topology of non-compact CY4 and the physicsof 3D $N=2$ theories in three different regimes. The first one is the Coulombbranch description when the CY4 is smooth. The second one is non-abelian gaugetheory when the CY4 has a degenerate $\mathbb{P}^1$-fibration structure. Thethird one is the strongly coupled SCFT from a CY4 singularity. We findinteresting flavor symmetry enhancements in the singular limit of CY4, as wellas an interesting and previously unexplored phenomenon in 3D, termed ``flavorsymmetry duality''. Many examples are analyzed with an emphasis on toric CY4sand $\mathbb{C}^4$ orbifolds with crepant resolutions. We develop a new branebox method to study the physics of Coulomb branch of 3D $N=2$ theory thatadmits a toric construction. Via IIB/M-theory duality we find that the branebox diagram living in $\mathbb{R}^3$ can be physically realized as aconfiguration of intersecting 4-branes which are extended objects in 8D maximalsupersymmetric theory, which is shown to be consistent via various chains ofdualities. The rank, effective gauge coupling and certain hints to flavorsymmetry enhancement of the 3D $N=2$ theory are read off from the brane box andcross-checked against the results obtained from geometric engineering. Theexotic branes in 8D maximal supersymmetric theory and the 4-string junctionsthereof are shown to play a crucial role in the construction of the brane box.

3d $N=2$ theories from M-theory on CY4 and IIB brane box