  Mutually unbiased bases for quantum degrees of freedom are central to all
theoretical investigations and practical exploitations of complementary
properties. Much is known about mutually unbiased bases, but there are also a
fair number of important questions that have not been answered in full as yet.
In particular, one can find maximal sets of ${N+1}$ mutually unbiased bases in
Hilbert spaces of prime-power dimension ${N=p^\m}$, with $p$ prime and $\m$ a
positive integer, and there is a continuum of mutually unbiased bases for a
continuous degree of freedom, such as motion along a line. But not a single
example of a maximal set is known if the dimension is another composite number
($N=6,10,12,...$). In this review, we present a unified approach in which the
basis states are labeled by numbers ${0,1,2,...,N-1}$ that are both elements of
a Galois field and ordinary integers. This dual nature permits a compact
systematic construction of maximal sets of mutually unbiased bases when they
are known to exist but throws no light on the open existence problem in other
cases. We show how to use the thus constructed mutually unbiased bases in
quantum-informatics applications, including dense coding, teleportation,
entanglement swapping, covariant cloning, and state tomography, all of which
rely on an explicit set of maximally entangled states (generalizations of the
familiar two--q-bit Bell states) that are related to the mutually unbiased
bases. There is a link to the mathematics of finite affine planes. We also
exploit the one-to-one correspondence between unbiased bases and the complex
Hadamard matrices that turn the bases into each other. The ultimate hope, not
yet fulfilled, is that open questions about mutually unbiased bases can be
related to open questions about Hadamard matrices or affine planes, in
particular the ...[rest deleted]
