An orientation D of a graph G=(V,E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v∈V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u)≠dD−(v), for all uv∈E(G). An orientation with maximum indegree at most k is called a k-orientation. The proper orientation number of G, denoted by χ→(G), is the minimum integer k such that G admits a proper k-orientation. We prove that determining whether χ→(G)≤k is NP-complete for chordal graphs of bounded diameter, but can be solved in linear-time in the subclass of quasi-threshold graphs. When parameterizing by k, we argue that this problem is FPT for chordal graphs and argue that no polynomial kernel exists, unless NP⊆coNP/poly. We present a better kernel to the subclass of split graphs and a linear kernel to the class of cobipartite graphs.Concerning bounds, we first prove that if G is split, then χ→(G)≤2ω(G)−2 and that if G is a k-uniform block graph, then χ→(G)≤3k−2. These bounds are tight. We also present new families of trees having proper orientation number at most 2 and at most 3. Actually, we prove a general bound stating that any graph G having no adjacent vertices of degree at least c+1 has proper orientation number at most c. This implies new classes of (outer)planar graphs with bounded proper orientation number. We also prove that maximal outerplanar graphs G whose weak-dual is a path satisfy χ→(G)≤13. Finally, we present simple bounds to the classes of chordal claw-free graphs and cographs.