The phase diagrams of systems described by a Hamiltonian containing an anisotropic quadratic term of the form $\frac{1}{2}g\ensuremath{\Sigma}{\ensuremath{\alpha}=1}^{n}{c}_{\ensuremath{\alpha}}\ensuremath{\int}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{}{S}_{\ensuremath{\alpha}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$, and a cubic anisotropic term $\ensuremath{\nu}\ensuremath{\Sigma}{\ensuremath{\alpha}=1}^{n}\ensuremath{\int}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{}{S}_{\ensuremath{\alpha}}^{4}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$, are studied using mean-field theory, scaling theory, and expansions in $\ensuremath{\epsilon}(=4\ensuremath{-}d)$ and $\frac{1}{n}$. Here, ${S}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ ($a=1, \dots{}, n$) is a local $n$-component ordering variable. Systems to which the analysis is applicable include perovskite crystals, stressed along the [100] direction ($n=3$), anisotropic antiferromagnets in a uniform field, uniaxially anisotropic ferromagnets, ferroelectric ferromagnets and crystalline $^{4}\mathrm{He}(n=2)$. When $g=0$ and $T={T}_{c}$ these systems undergo a phase transition that may be associated (for small $n$) with the Heisenberg fixed point (${\ensuremath{\nu}}^{*}=0$) or (otherwise) with the cubic fixed point (${\ensuremath{\nu}}^{*}>0$) of the renormalization group. Although $\ensuremath{\nu}$ is an "irrelevant variable" in the former case, it is found to have important effects. For $\ensuremath{\nu}<0$, the point $g=0$, $T={T}_{c}$ represents a bicritical point in the $g\ensuremath{-}T$ plane, at which a first-order "spin-flop" line (separating two distinct ordered phases) meets two critical lines. For $\ensuremath{\nu}>0$, the "flop" line splits into two critical lines, associated with transitions between each of the ordered phases and a new intermediate phase; the point $T={T}_{c}$, $g=0$ is then tetracritical. The shape of the boundary of the intermediate phase is given by $T={T}_{2}(g, \ensuremath{\nu})$ with $[{T}_{c}\ensuremath{-}{T}_{2}(g, \ensuremath{\nu})]\ensuremath{\sim}{(\frac{g}{\ensuremath{\nu}})}^{\frac{1}{{\ensuremath{\psi}}_{2}}}$, where ${\ensuremath{\psi}}_{2}={\ensuremath{\varphi}}_{g}\ensuremath{-}{\ensuremath{\varphi}}_{\ensuremath{\nu}}$ (if the tetracritical point is Heisenberg-like) or ${\ensuremath{\psi}}_{2}={\ensuremath{\varphi}}_{g}^{C}$ (if it is cubic). Here, ${\ensuremath{\varphi}}_{g}$, ${\ensuremath{\varphi}}_{\ensuremath{\nu}}$, and ${\ensuremath{\varphi}}_{g}^{C}$ are appropriate crossover exponents associated with the two symmetry-breaking perturbations. The phase diagram of [111] -stressed perovskites is also discussed and the experimental situation briefly reviewed.