Abstract: Design theory has various applications in other fields including coding theory, statistics, computer science and cryptography. For example, 2-designs have been used from the early years of this century for optimal planning of statistical experiments. This dissertation contributes a method of creating new 3-designs. A $t$-$(v,k,\lambda)$ design is a pair $(X,{\mathcal{A}})$, where $X$ is a set of $v$ elements (called points) and a family $\mathcal{A}$ of subsets of $X$, each of size $k$ (called blocks) such that each $t$-element subset of points of $X$ is contained in exactly $\lambda$ blocks. A $t$-$(v,k,\lambda)$ design is also denoted by $S_{\lambda}(t,k,v)$. The problem of finding all parameters $(t,v,k, \lambda)$ for which $t$-designs exist is a long standing unsolved problem. It has been proved that if $\lambda$ is large and the parameters $(t,v,k,\lambda)$ satisfy the necessary arithmetic conditions then there exists a $t$-$(v,k,\lambda)$ design. If $\lambda =1$, a $t$-$(v,k,1)$ design is called a Steiner $t$-design (or Steiner system) and is also denoted by $S(t,k,v)$. Steiner designs are particularly important in application problems like statistical experiments and network designs where they are used to minimize construction costs. There are many known Steiner 2-designs, but constructing Steiner $t$-designs with $t>2$ is much harder. Problems that are of interest in design theory are (1) determining all quadruples $(t,v,k,\lambda)$ for which a $t$-$(v,k,\lambda)$ design exists (the existence problem), (2) economical construction of the blocks of a $t$-$(v,k,\lambda)$ design (the construction problem), (3) finding $t$-designs with large automorphism groups (the symmetry problem) and (4) enumerating the number of distinct $t$-$(v,k,\lambda)$ designs on a given point set of size $v$ (the enumeration problem). This dissertation contributes to a solution of the existence problem for $t>2$, particularly for $t=3$ and $\lambda =1$ (the Steiner 3-designs). Combinatorial structures called candelabra systems can be used in recursive constructions to build new Steiner 3-designs. We introduce a new closure operation on natural numbers involving candelabra systems. Many of the constructions of Steiner designs are based on the ``block spreading'' construction theorem. The generalization of this theorem is presented in this dissertation. This new closure operation and the generalized ``block spreading'' construction make it possible to generalize various constructions for Steiner $3$-designs and to create new infinite families of Steiner 3-designs as follows: let $q$ be a prime power. Assume that there is a Steiner 3-design 3-$(a+1,q+1,1)$, $a>1$. For every $v$ satisfying certain necessary arithmetic conditions we can construct a Steiner 3-design 3-$(va^d+1,q+1,1)$ for every $d$ sufficiently large. In the case of block size 6 this theorem yields new infinite families of Steiner 3-designs: if $v$ is a given positive integer satisfying certain arithmetic conditions, for every non-negative integer $m$ there exists a Steiner 3-design 3-$(v{(4\cdot 5^m+1)}^{\textstyle d}+1),6,1)$ for sufficiently large $d$. This new theory produces new Steiner 2-designs as well: If $q$ and $q+1$ are both prime powers the existence of the Steiner 2-designs 2-$(a+1,q+1,1)$ and 2-$(b+1,q+1,1)$ implies the existence of a Steiner 2-design 2-$(ab+b+1,q+1,1)$.