Lesbian couples or individuals have to go through a little more trouble, and gay couples and individuals have to jump through a considerable number of hoops. If you're asexual as a small percentage of people in every country are trying to conceive might just be really tough. What options do asexual people who would like to become parents have?
Asexual women and trying to conceive
Asexual women who would like to get pregnant have more options than asexual men have. Some have sex for the purpose of reproduction despite the fact that they do not experience sexual feelings, but others find the sensation of sex to be utterly repulsive and could not engage in sexual intercourse even in order to get pregnant.
These women could, theoretically speaking, choose to use one of many artificial insemination techniques. The most simple of these techniques is trans-vaginal insemination also known as "DIY insemination" if you do it yourself. A doctor can help you with this technique, bit it's really very simple. The sperm sample is sucked into the (needle-less!) syringe you'll be using, and then discharged into the vagina around the time of ovulation. Women who have male partners will probably want to use their partners' sperm, while single women can use a sperm bank or find a donor on their own.
Those who are looking for a known donor among their friends and acquaintances will want to ask candidates to submit to blood and STD tests before moving ahead. Intrauterine insemination, or IUI for short, is an artificial reproductive technique that is a little more involved. The sperm you'll be inseminated with needs to go through a sperm washing process before it can be inserted into your uterus. You will also usually be given fertility drugs to make the timing of your ovulation more predictable. There is no need for asexual women to use IVF to get pregnant, unless they have a medical problem such as endometriosis or blocked fallopian tubes. IVF is pricey and not the first fertility treatment you want to turn to if you need artificial insemination for non-medical reasons.
Asexual men who would like to become fathers
The options available to asexual men who want to become fathers tend to be more limited. Some are both willing and able to have sexual intercourse for reproductive purposes. Some aren't, and of those some are able and willing to produce sperm samples through masturbation. Men in the latter category should talk to their doctors about the best way to get their partner or a surrogate pregnant. If sperm samples are going to be readily available, you'll then just have to choose an insemination technique.
Once again, trans-vaginal insemination is the simplest way to go. Intrauterine insemination and invitro fertilization are other options that could be best, depending on the situation. Men who are using an egg donor and surrogate mother to conceive clearly need to go the IVF way. Regulations covering egg donors and surrogate mothers vary greatly from jurisdiction to jurisdiction. Making sure you understand all the possible legal implications is a much bigger job than the IVF itself, and I'd advise you to get a lawyer experienced in fertility issues on board before starting the process.
Choosing your egg donor and surrogate is a very important process, legally as well as emotionally and practically. Asexual men who can't or won't have sex or masturbate practically have only one option, and that is to have sperm extracted directly from their testicles. The sperm sample that is obtained can then be used in a procedure called intracytoplasmic sperm injection (ICSI), where one sperm is isolated and directly injected into a single egg.
This procedure is a sub-category of IVF. Asexual people of both genders, partnered or not, can also consider foster parenting and adoption. Married couples (sexual or asexual!) have a relatively easy time doing this, though adoption can be very pricey especially international adoption. Fewer countries are willing to consider individuals who are not married or partnered as adoptive parents, but there are some and it is worth doing the research. Why did we call the insemination options "theoretical"? As with adoption, artificial reproductive techniques can be very, very expensive. Not everyone has the money to pay for these procedures out of pocket or even with a payment plan (often offered by fertility clinics). Insurance policies sometimes cover artificial reproductive techniques, but even if yours does they may not be available to you unless you have a proven medical problem that causes infertility. In practical terms, starting with the simplest possible step and working your way up from there is the best.