To what extent do Mormons believe that miraculous healing happens in our days? Gospel accounts about Jesus' life make it seem like he miraculously healed large amounts of people during the time he was here on earth. For example, Matthew 4:23,24 (KJV) says this:


  And Jesus went about all Galilee, teaching in their synagogues, and
  preaching the gospel of the kingdom, and healing all manner of
  sickness and all manner of disease among the people.
  
  And his fame went throughout all Syria: and they brought unto him all
  sick people that were taken with divers diseases and torments, and
  those which were possessed with devils, and those which were lunatick,
  and those that had the palsy; and he healed them.


Reading above passage it seems like Jesus' miracles provided a form of advertising about him, since verse 24 speaks of his "fame" that led people with all kinds of illnesses to be brought to him.

There are at least some Christian denominations that believe that God also heals people during our times, but if this was actually the case, we probably wouldn't have such an overburdened health care system.

I mean no doubt most people would probably rather announce having faith in Jesus than pay expensive fees for medical treatment. Even though above passage actually makes it look like Jesus also healed people who were not really worshipers of God, such as those that were "possessed with devils".

Anyway, since we don't see any Christian denominations reducing the backlogs in our hospitals nowadays, I wanted to ask about the extent to which different denominations believe in miraculous physical healing happening in our time.

For example, I saw that the seventh Article of Faith published at lds.org says:


  We believe in the gift of tongues, prophecy, revelation, visions,
  healing, interpretation of tongues, and so forth.


(emphasis added)

Do Mormons believe that people are physically healed in our times or that this refers to spiritual healing, or to something that is expected to happen in the future?
Yes, Mormons believe in gifts and miracles of both spiritual and physical healing in these days.

Elder Dallin H. Oaks of the Quorum of the Twelve Apostles gave a talk in 2010 called Healing the Sick. The whole talk is relevant to your question but here's a snippet (not sorry for making it essentially my whole answer):


  Latter-day Saints believe in applying the best available scientific knowledge and techniques. We use nutrition, exercise, and other practices to preserve health, and we enlist the help of healing practitioners, such as physicians and surgeons, to restore health.
  
  The use of medical science is not at odds with our prayers of faith and our reliance on priesthood blessings. When a person requested a priesthood blessing, Brigham Young would ask, “Have you used any remedies?” To those who said no because “we wish the Elders to lay hands upon us, and we have faith that we shall be healed,” President Young replied: “That is very inconsistent according to my faith. If we are sick, and ask the Lord to heal us, and to do all for us that is necessary to be done, according to my understanding of the Gospel of salvation, I might as well ask the Lord to cause my wheat and corn to grow, without my plowing the ground and casting in the seed. It appears consistent to me to apply every remedy that comes within the range of my knowledge, and [then] to ask my Father in Heaven … to sanctify that application to the healing of my body.”
  
  Of course we don’t wait until all other methods are exhausted before we pray in faith or give priesthood blessings for healing. In emergencies, prayers and blessings come first. Most often we pursue all efforts simultaneously. This follows the scriptural teachings that we should “pray always” (D&amp;C 90:24) and that all things should be done in wisdom and order.
  
  ...
  
  Truly, as the Book of Mormon teaches, God “manifesteth himself unto all those who believe in him, by the power of the Holy Ghost; yea, unto every nation, kindred, tongue, and people, working mighty miracles … among the children of men according to their faith” (2 Nephi 26:13).
  
  ...
  
  Many scriptures teach that the servants of the Lord “shall lay hands on the sick, and they shall recover” (Mark 16:18). Miracles happen when the authority of the priesthood is used to bless the sick. I have experienced these miracles. As a boy and as a man I have seen healings as miraculous as any recorded in the scriptures, and so have many of you.


You might say that the LDS believe in modern medicine as a miracle in and of itself. Elder Russell M. Nelson, another member of the Twelve, is a world-renowned heart surgeon who developed several new techniques for saving lives. But, as he has stated in his own talks on the matter, not without faith and prayers and fasting.

The Book of Mormon says, in Moroni 7:27,29,35-36:


  27 Wherefore, my beloved brethren, have miracles ceased because Christ hath ascended into heaven, and hath sat down on the right hand of God, to claim of the Father his rights of mercy which he hath upon the children of men?
  
  29 And because he hath done this, my beloved brethren, have miracles ceased? Behold I say unto you, Nay; neither have angels ceased to minister unto the children of men.
  
  35 And now, my beloved brethren, if this be the case that these things are true which I have spoken unto you, and God will show unto you, with power and great glory at the last day, that they are true, and if they are true has the day of miracles ceased?
  
  36 Or have angels ceased to appear unto the children of men? Or has he withheld the power of the Holy Ghost from them? Or will he, so long as time shall last, or the earth shall stand, or there shall be one man upon the face thereof to be saved?
  
  37 Behold I say unto you, Nay; for it is by faith that miracles are wrought; and it is by faith that angels appear and minister unto men; wherefore, if these things have ceased wo be unto the children of men, for it is because of unbelief, and all is vain.