There are three underlying classes of algebraic varieties: General Type varieties, Calabi-Yau varieties, and Fano varieties. One of the key goals in complex geometry is to study when the above classes of varieties admit canonical metrics. An important example of such metrics is the Kähler-Einstein metrics. General type and Calabi-Yau varieties always admit a unique Kähler-Einstein metric. Fano varieties, however, are more complicated since it is known that some Fano varieties do not admit a Kähler-Einstein metric. To prove that varieties are K-stable it is possible to compute stability thresholds (δ-invariants). Generalizing this concept, one can also consider log Fano pairs so that stability conditions will tell us if there is a Kähler-Einstein metric on the regular locus of a variety with volume equal to the algebraic volume. In my talk, I will discuss log Fano planes, meaning that I will consider curves on a projective plane for degrees 1, 2, 3, and 4. I will describe a classification of such curves and present a computation of δ-invariant for one of the curves as an example.  I will also summarize my results and show possible applications.