The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are: 1. Every 3-connected simple planar graph G that contains a k-path, a path on k vertices, also contains a k-path P such that for its weight (the sum of degrees of its vertices) in G it holds $w_G\left(P\right):={\sum }_{u\in V\left(P\right)}de{g}_G\left(u\right)\le (3/2)k²+\left(k\right)$ 2. Every plane triangulation T that contains a k-path also contains a k-path P such that for its weight in T it holds $w_T\left(P\right):={\sum }_{u\in V\left(P\right)}de{g}_T\left(u\right)\le k²+13k$ 3. Let G be a 3-connected simple planar graph of circumference...