Let G be a simple graph of order n with no isolated edges and at most one isolated vertex. For a positive integer w, a w-weighting of G is a function $f:E(G)\rightarrow\{1,2,\dots,w\}$. An irregularity strength of G, $s(G)$, is the smallest w such that there is a w-weighting of G for which $\sum_{e:u\in e}f(e)\neq\sum_{e:v\in e}f(e)$ for all pairs of different vertices $u,v\in V(G)$. We prove that $s(G)<112\frac{n}{\delta}+28$, where $\delta$ is the minimum degree of G. For d-regular graphs, we strengthen this to $s(G)<40\frac{n}{d}+11$. These upper bounds represent improvements of many existing ones. Similar results concerning the “total” version of the irregularity strength are also discussed.