  Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal
of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept
of directed strongly regular graphs. In this paper we construct several rich
families of directed strongly regular graphs with new parameters. Our
constructions yielding new parameters are based on extending known explicit
constructions to cover more parameter sets. We also explore some of the links
between Cayley graphs, block matrices and directed strongly regular graphs with
certain parameters. Directed strongly regular graphs which are also Cayley
graphs are interesting due to their having more algebraic structure. We
construct directed strongly regular Cayley graphs with parameters
$((m+1)s,ls,ld,ld-d,ld)$ where $d,l$ and $s$ are integers with $dm=ls$ and
$1\leq l<m$. We also give a new block matrix characterization for directed
strongly regular graphs with parameters $(m(dm+1),dm,m,m-1,m)$, which were
first dicussed by Duval et al. in "Semidirect Product Constructions of Directed
Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 104
(2003) 157 - 167].
