  Although the basic idea behind the concept of a greedy basis had been around
for some time, the formal development of a theory of greedy bases was initiated
in 1999 with the publication of the article [S.~V.~Konyagin and
V.~N.~Temlyakov, A remark on greedy approximation in Banach spaces, East J.
Approx. 5 (1999), no. 3, 365--379]. The theoretical simplicity of the
thresholding greedy algorithm became a model for a procedure widely used in
numerical applications and the subject of greedy bases evolved very rapidly
from the point of view of approximation theory. The idea of studying greedy
bases and related greedy algorithms attracted also the attention of researchers
with a classical Banach space theory background. From the more abstract point
of functional analysis, the theory of greedy bases and its derivates evolved
very fast as many fundamental results were discovered and new ramifications
branched out. Hundreds of papers on greedy-like bases and several monographs
have been written since the foundational paper mentioned above appeared. After
twenty-five years, the theory is very much alive and it continues to be a very
active research topic both for functional analysts and for researchers
interested in the applied nature of nonlinear approximation alike. This is why
we believe it is a good moment to gather a selection of 25 open problems (one
per year since 1999!) whose solution would contribute to advance the state of
art of this beautiful topic.
