  We review the current status of one dimensional periodic potentials and also
present several new results. It is shown that using the formalism of
supersymmetric quantum mechanics, one can considerably enlarge the limited
class of analytically solvable one-dimensional periodic potentials. Further,
using the Landen transformations as well as cyclic identities for Jacobi
elliptic functions discovered by us recently, it is shown that a linear
superposition of Lam\'e (as well as associated Lam\'e) potentials are also
analytically solvable. Finally, using anti-isospectral transformations, we also
obtain a class of analytically solvable, complex, PT-invariant, periodic
potentials having real band spectra.
