  The work is devoted to the variety of $2$-dimensional algebras over an
algebraically closed field. Firstly, we classify such algebras modulo
isomorphism. Then we describe the degenerations and the closures of principal
algebra series in the variety under consideration. Finally, we apply our
results to obtain analogous descriptions for the subvarieties of flexible, and
bicommutative algebras. In particular, we describe rigid algebras and
irreducible components for these subvarieties.
