  Starting from a multichannel quantum-defect theory, we derive analytic
descriptions of a magnetic Feshbach resonance in an arbitrary partial wave $l$,
and the atomic interactions around it. An analytic formula, applicable to both
broad and narrow resonances of arbitrary $l$, is presented for ultracold atomic
scattering around a Feshbach resonance. Other related issues addressed include
(a) the parametrization of a magnetic Feshbach resonance of arbitrary $l$, (b)
rigorous definitions of "broad" and "narrow" resonances of arbitrary $l$ and
their different scattering characteristics, and (c) the tuning of the effective
range and the generalized effective range by a magnetic field.
